Investigation of the two-cut phase region in the complex cubic ensemble of random matrices

Barhoumi, Ahmad; Bleher, Pavel; Deaño, Alfredo; Yattselev, Maxim

doi:10.1063/5.0086911

We investigate the phase diagram of the complex cubic unitary ensemble of random matrices with the potential $V (M) = - \frac{1}{3} M^{3} + t M$ ⁠, where t is a complex parameter. As proven in our previous paper [Bleher et al., J. Stat. Phys. 166, 784–827 (2017)], the whole phase space of the model, $t \in C$ ⁠, is partitioned into two phase regions, $O_{one−cut}$ and $O_{two−cut}$ ⁠, such that in $O_{one−cut}$ the equilibrium measure is supported by one Jordan arc (cut) and in $O_{two−cut}$ by two cuts. The regions $O_{one−cut}$ and $O_{two−cut}$ are separated by critical curves, which can be calculated in terms of critical trajectories of an auxiliary quadratic differential. In Bleher et al. [J. Stat. Phys. 166, 784–827 (2017)], the one-cut phase region was investigated in detail. In the present paper, we investigate the two-cut region. We prove that in the two-cut region, the endpoints of the cuts are analytic functions of the real and imaginary parts of the parameter t, but not of the parameter t itself (so that the Cauchy–Riemann equations are violated for the endpoints). We also obtain the semiclassical asymptotics of the orthogonal polynomials associated with the ensemble of random matrices and their recurrence coefficients. The proofs are based on the Riemann–Hilbert approach to semiclassical asymptotics of the orthogonal polynomials and the theory of S-curves and quadratic differentials.

I. INTRODUCTION

This work is a continuation of the works of Bleher and Deaño^1,2 and Bleher, Deaño, and Yattselev.³ The main goal of the whole project is to analyze the large N asymptotics of the partition function and correlation functions in the unitary ensemble of random matrices with the cubic potential $V (M) = - \frac{1}{3} M^{3} + t M$ ⁠, where t is a complex parameter. The partition function of this ensemble is formally defined by the matrix integral over the space of N × N Hermitian matrices,

Z_{N}^{RM} (t) = \int_{H_{N}} e^{- N T r (- \frac{1}{3} M^{3} + t M)} d M .

The integral is formal because it is divergent, and it needs a regularization. To define such a regularization, consider the (formal) identity

Z_{N}^{RM} (t) = \frac{π^{N (N - 1) / 2}}{\prod_{k = 1}^{N} k!} \int_{R} \dots \int_{R} \prod_{1 \leq j < k \leq N} {(z_{j} - z_{k})}^{2} \prod_{j = 1}^{N} e^{- N (- \frac{1}{3} z_{j}^{3} + t z_{j})} {d z}_{1} \dots {d z}_{N}

{see Ref. 4 [formula (1.2.18)]}. The random matrix partition function $Z_{N}^{RM} (t)$ can now be regularized by changing the contour of integration from the real line $R$ to the contour Γ depicted in Fig. 1, which goes from (−∞) to (e^πi/3∞). Hence, we are interested in studying the regularized eigenvalue partition function defined as

Z_{N} (t) ≔ \int_{Γ} \dots \int_{Γ} \prod_{1 \leq j < k \leq N} {(z_{j} - z_{k})}^{2} \prod_{j = 1}^{N} e^{- N (- \frac{1}{3} z_{j}^{3} + t z_{j})} {d z}_{1} \dots {d z}_{N},

(1.1)

which is well defined (the integrals are convergent) for all values $t \in C$ ⁠.

FIG. 1.

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The contour Γ of integration.

As shown in Ref. 2, the partition function Z_N(t) can be used to study the topological expansion in the cubic ensemble of random matrices and enumeration of regular graphs of degree 3 on Riemannian surfaces. To be more precise, let us consider the cubic potential in the form M²/2 − uM³, where u > 0, and the corresponding eigenvalue matrix integral is given by

Ξ_{N} (u) = \int_{Γ} \dots \int_{Γ} \prod_{1 \leq j < k \leq N} {(ζ_{j} - ζ_{k})}^{2} \prod_{j = 1}^{N} e^{- N (\frac{1}{2} ζ_{j}^{2} - u ζ_{j}^{3})} d ζ_{1} \dots d ζ_{N} .

(1.2)

We would like to consider all the possible values, including the complex ones, of the parameter u. Formula (1.2) is not very convenient for this purpose because the contour of integration Γ should be rotated to secure the convergence of the integral depending on the argument of u. Instead, let us make the change of variables

ζ_{j} = {(3 u)}^{- 1 / 3} z_{j} + \frac{1}{6 u},

which yields that

\frac{ζ_{j}^{2}}{2} - u ζ_{j}^{3} - \frac{1}{108 u^{2}} = - \frac{z_{j}^{3}}{3} + t z_{j},

where

t = \frac{1}{4 {(3 u)}^{4 / 3}} .

(1.3)

This implies the relation between the partition functions,

Ξ_{N} (u) = {(4 t)}^{N^{2} / 4} e^{- 2 t^{3 / 2} N / 3} Z_{N} (t) .

As proven in Ref. 2, the free energy of the cubic model $F_{N} (u)$ admits an asymptotic expansion as N → ∞ in powers of N⁻², that is,

F_{N} (u) ≔ \frac{1}{N^{2}} \ln \frac{Ξ_{N} (u)}{Ξ_{N} (0)} \sim \sum_{g = 0}^{\infty} \frac{F^{(2 g)} (u)}{N^{2 g}}

(1.4)

for any u in the interval 0 ≤ u < u_c, where u_c ≔ 3^1/418⁻¹ is a critical point. In addition, the functions F^(2g)(u) admit an analytic continuation to the disk |u| < u_c on the complex plane, and if we expand them in powers of u,

F^{(2 g)} (u) = \sum_{j = 1}^{\infty} \frac{f_{2 j}^{(2 g)} u^{2 j}}{(2 j)!},

(1.5)

then the coefficient $f_{2 j}^{(2 g)}$ is a positive integer that counts the number of 3-valent connected graphs with 2j vertices on a Riemann surface of genus g. Asymptotic expansion (1.4) is called a topological expansion. For more details on this aspect of the theory, we refer the reader to the classical papers;^5,6 the monograph of Forrester⁷ (Sec. 1.6); and the works of Mulase,⁸ Di Francesco,⁹ Ercolani and McLaughlin,^10,11 Eynard,¹² and references therein, or a very readable introduction by Zvonkin.¹³ It is noteworthy that the general idea of a topological expansion goes back to the classical work of t’ Hooft.¹⁴

As shown in Ref. 2, the coefficients $f_{2 j}^{(2 g)} / (2 j)!$ of power series (1.5) behave, when j → ∞, as

\frac{f_{2 j}^{(2 g)}}{(2 j)!} = \frac{K_{2 g} j^{\frac{5 g - 7}{2}}}{u_{c}^{2 j}} (1 + O (j^{- 1 / 2})), K_{2 g} > 0 .

This implies that u_c is the radius of convergence of each power series (1.5). In fact, u = u_c is a singular point of each of the functions (1.5). The topological expansion in a neighborhood of the critical point u_c has been obtained in the work of Bleher and Deaño,¹ and it is closely related to the Painlevé I equation as follows: consider a formal series

\sum_{g = 0}^{\infty} Γ (\frac{5 g - 1}{2}) \frac{u_{c}^{g} K_{2 g}}{6 \cdot 3^{1 / 4}} λ^{\frac{1 - 5 g}{2}},

(1.6)

and it can be shown that there exists a one-parameter family of solutions {y_α(λ)}, $α \in R$ ⁠, of the Painlevé I differential equation

y^{″} (λ) = a_{0} y^{2} (λ) - a_{1} λ,

with $a_{0} = 2^{\frac{5}{2}} 3^{\frac{9}{4}}$ and $a_{1} = 2^{\frac{3}{2}} 3^{- \frac{5}{4}}$ ⁠, that admits an asymptotic expansion (1.6) as λ → −∞. This family consists of the Boutroux tronquée solutions, which means that they are asymptotically free of poles in one of the five canonical sectors of angle 2π/5 in the complex plane that appear naturally when considering rotational symmetries of the Painlevé I equation. The parameter α appears in the Painlevé I Riemann–Hilbert problem, as described by Kapaev in Ref. 15, and in the context of the cubic model, it is related to the choice of the original contour of integration Γ (see Refs. 1 and 2). In the present case, we have α = 1, which makes the solution of Painlevé I tritronquée, that is, asymptotically free of poles in four of the five canonical sectors of angle 2π/5 in the complex plane. We refer to the reader to Ref. 1 (Sec. 2.1) and also the work of Joshi and Kitaev¹⁶ for more details.

It is noteworthy that the key ingredient in the proof of topological expansion (1.4) in Ref. 8 is the derivation of semiclassical asymptotic formulas for the recurrence coefficients $γ_{n}^{2}$ and β_n [see (1.13), where the parameter t is related to u via (1.3)] of the corresponding monic orthogonal polynomials P_n(z) = zⁿ + ⋯ defined via orthogonality relations

\int_{Γ} P_{n} (z) z^{k} e^{- N (\frac{1}{2} z^{2} - u z^{3})} d z = 0, k \in {0, 1, \dots, n - 1} .

The idea of using the orthogonal polynomials in ensembles of random matrices goes back to the classical works of Dyson and Mehta (see Refs. 17 and 18 and references therein). In particular, as proven in Ref. 2, for any u ∈ [0, u_c), there exists ɛ > 0 such that as N, n → ∞ with $1 - ε \leq \frac{n}{N} \leq 1 + ε$ ⁠, the recurrence coefficients $γ_{n}^{2}$ and β_n admit asymptotic expansions in powers of $\frac{1}{N^{2}}$ ⁠,

\{\begin{aligned} γ_{n}^{2} & \sim \sum_{k = 0}^{\infty} \frac{1}{N^{2 k}} g_{2 k} (\frac{n}{N}, u), \\ β_{n} & \sim \sum_{k = 0}^{\infty} \frac{1}{N^{2 k}} b_{2 k} (\frac{n}{N} + \frac{1}{2 N}, u), \end{aligned})

(1.7)

where the functions g_2k(s, u), b_2k(s, u) do not depend on n and N and are analytic in s at s = 1.

In Ref. 1, this asymptotic expansion is extended to the double scaling asymptotic expansion of the recurrence coefficients at the critical point u_c. In the double scaling regime, we set

\frac{n}{N} = 1 + v N^{- 4 / 5},

where $v \in R$ is a scaling variable. Then, as proven in Ref. 1, at u = u_c, the recurrence coefficients $γ_{n}^{2}$ and β_n admit asymptotic expansions in powers of N^−2/5 as N → ∞,

\{\begin{aligned} γ_{n}^{2} & \sim γ_{c}^{2} + \sum_{k = 1}^{\infty} \frac{1}{N^{2 k / 5}} p_{2 k} (v), \\ β_{n} & \sim β_{c} + \sum_{k = 1}^{\infty} \frac{1}{N^{2 k / 5}} q_{2 k} (\tilde{v}), \end{aligned})

(1.8)

where the functions p_2k(v), $q_{2 k} (\tilde{v})$ are expressed in terms of the Boutroux tritronquée solution to the Painlevé I equation, $γ_{c}^{2}, β_{c}$ are known explicitly, and $\tilde{v} = v + N^{- 1 / 5} / 2$ ⁠. As shown in Ref. 1, expansions (1.7) and (1.8) can be extended for large N to u in overlapping intervals: [0, u_c − N^−0.79] for (1.7) and [u_c − N^−0.65, u_c] for (1.8), and this can be used to obtain the double scaling asymptotic formula for the partition function at u_c. In particular, let

u - u_{c} = 2^{- \frac{12}{5}} 3^{- \frac{7}{4}} λ N^{- \frac{4}{5}},

where λ is a complex scaling variable in the double scaling regime. Then, for λ outside of a neighborhood of the poles of y(λ), the Boutroux tritronquée solution to Painlevé I equation, the partition function Ξ_N(u) can be written as

Ξ_{N} (u) = Ξ_{N}^{reg} (u) Ξ_{N}^{sing} (λ) (1 + O (N^{- ε})), ε > 0,

(1.9)

where the regular and singular factors are given by

Ξ_{N}^{reg} (u) = e^{N^{2} [a + b (u - u_{c}) + c {(u - u_{c})}^{2}] + d} and Ξ_{N}^{sing} (λ) = e^{- Y (λ)},

where a, b, c, d are some explicit constants and Y(λ) is a solution of the differential equation Y″(λ) = y(λ) with the boundary condition

Y (λ) = \frac{2 \sqrt{6}}{45} {(- λ)}^{5 / 2} - \frac{1}{48} \log (- λ) + O ({(- λ)}^{- 5 / 2}), λ \to - \infty .

Asymptotic formula (1.9) is used in Ref. 1 to prove the conjecture of David^19,20 that the poles of the tritronquée solution y(λ) give rise to zeros of Ξ_N(u).

As we have stressed before, the topological expansions from Refs. 1–3 were obtained by first analyzing the associated orthogonal polynomials defined by

\int_{Γ} z^{k} P_{n} (z; t, N) e^{- N V (z; t)} d z = 0, k \in {0, \dots, n - 1},

(1.10)

where

V (z; t) ≔ - \frac{z^{3}}{3} + t z, t \in C .

(1.11)

Due to the non-Hermitian character of the above relations, it might happen that polynomial satisfying (1.10) is non-unique. In this case, we understand by P_n(z; t, N) the monic non-identically zero polynomial of the smallest degree (such a polynomial is always unique). One way of connecting the partition function to orthogonal polynomials is via three term recurrence relations. More precisely, let

h_{n} (t, N) ≔ \int_{Γ} P_{n}^{2} (z; t, N) e^{- N V (z; t)} d z = \frac{D_{n} (t, N)}{D_{n - 1} (t, N)},

(1.12)

where D₋₁(t, N) ≡ 1 and $D_{n} (t, N) ≔ d e t {[\int_{Γ} z^{i + j} e^{- N V (z; t)} d z]}_{i, j = 0}^{n}$ is the Hankel determinant of the moments of the measure of integration in (1.10). It easy to see that each D_n(t, N) is an entire function of t, and therefore, each h_n(t, N) is meromorphic in $C$ ⁠. Hence, given n, the set of the values t for which there exists k ∈ {0, …, n} such that h_k(t, N) = 0 is countable with no limit points in the finite plane. Outside of this set, the standard argument using orthogonality (1.10) shows that

z P_{n} (z; t, N) = P_{n + 1} (z; t, N) + β_{n} (t, N) P_{n} (z; t, N) + γ_{n}^{2} (t, N) P_{n - 1} (z; t, N),

(1.13)

and by analytic continuation, (1.13) extends to those values of t for which D_n−1(t, N)D_n(t, N) ≠ 0 [that is, n + 1-th and nth polynomials appearing (1.13) have the prescribed degrees], where

γ_{n}^{2} (t, N) = h_{n} (t, N) / h_{n - 1} (t, N) .

(1.14)

Note that (1.13) remains meaningful even if D_n−2(t, N) = 0. In this case, h_n−2(t, N) = 0, h_n−1(t, N) = ∞, and h_n(t, N) is finite, which means that P_n−2(z; t, N) is orthogonal to itself and, therefore, is equal to P_n−1(z; t, N) but not to P_n(z; t, N). Thus, $γ_{n}^{2} (t, N) = 0$ ⁠, and the last term in (1.13) is zero. Note also that if h_n−1(t, N) ≠ 0 while h_n(t, N) = 0 [that is, D_n−1(t, N) ≠ 0 and D_n(t, N) = 0], then we simply have that $γ_{N}^{2} (t, N) = 0$ by analytic continuation. However, this means that the polynomial P_n(z; t, N) is orthogonal to itself and, therefore, P_n+1(z; t, N) = P_n(z; t, N). Then, the analytic continuation argument necessitates that β_n(t, N) = ∞, which also can be seen from the determinantal representation of β_n(t, N) as it is the difference of the subleading coefficients of P_n(z; t, N) and P_n+1(z; t, N). It is further known that the recurrence coefficients $γ_{N}^{2} (t, N)$ satisfy the Toda equation,

\frac{\partial^{2} F_{N} (t)}{\partial t^{2}} = γ_{N}^{2} (t, N), F_{N} (t) = \frac{1}{N^{2}} \log Z_{N} (t) .

Another way of connecting the partition function to orthogonal polynomials is through the formula

Z_{N} (t) = N! \prod_{n = 0}^{N - 1} h_{n} (t, N) (= N! D_{N - 1} (t, N)) .

As it turns out, there are two different regions in the complex t-plane for which the behavior of the polynomials P_n(z; t, N), and therefore of the partition function, is different. Colloquially, we call them one-cut and two-cut regions, a distinction that will become clear later. In Refs. 1–3, the partition function has been analyzed in and on the boundary of the one-cut region. The goal of this work is to start the analysis in the two-cut region. More precisely, here we only consider the asymptotics of the orthogonal polynomials and their recurrence coefficients and postpone the analysis of the partition function for the future project. The structure of this paper is as follows:

In Secs. II and III, we describe equilibrium measures and corresponding S-curves for the cubic model under consideration (those describe asymptotic behavior of the normalized counting measures of zeros of the orthogonal polynomials in the weak * sense). This leads us to a precise description of the phase diagram (i.e., of the one- and two-cut regions) of the cubic model on the complex t-plane.
In Sec. IV, we present the main results of this paper: asymptotic formulas for the orthogonal polynomials and their recurrence coefficients in the two-cut phase region.
Sections V–IX are devoted to the proof of our main results; in them, we obtain various results about the detailed structure of the S-curves, derive main properties of the dominant terms of the expansions, apply the Riemann–Hilbert approach to asymptotics of the orthogonal polynomials, and finally prove the expansions themselves.

II. EQUILIBRIUM MEASURES AND S-PROPERTY

It is well understood that the zeros of polynomials satisfying (1.10) asymptotically distribute as a certain weighted equilibrium measure on an S-contour corresponding to the weight function (1.11). In this section, we discuss these notions in greater detail. Our consideration will use the works of Huybrechs, Kuijlaars, and Lejon²¹ and Kuijlaars and Silva.²² Let us start with some definitions.

Definition 2.1.

Let V be an entire function. The logarithmic energy in the external field ReV of a measure ν in the complex plane is defined as

E_{V} (ν) = \iint \log \frac{1}{| s - t |} d ν (s) d ν (t) + \int R e V (s) d ν (s) .

The equilibrium energy of a contour Γ in the external field ReV is equal to

E_{V} (Γ) = \inf_{ν \in M (Γ)} E_{V} (ν),

(2.1)

where

M (Γ)

denotes the space of Borel probability measures on Γ.

When ReV(s) − log|s| → +∞ as Γ ∋ s → ∞, there exists a unique minimizing measure for (2.1), which is called the weighted equilibrium measure of Γ in the external field ReV, say, μ_Γ [see Ref. 23 (Theorem I.1.3) or Ref. 21]. The support of μ_Γ, say, J_Γ, is a compact subset of Γ. The equilibrium measure μ = μ_Γ is characterized by the Euler–Lagrange variational conditions,

2 U^{μ} (z) + R e V (z) \{\begin{aligned} = ℓ, z \in J_{Γ}, \\ \geq ℓ, z \in Γ \ J_{Γ}, \end{aligned})

(2.2)

where ℓ = ℓ_Γ is a constant, the Lagrange multiplier, and U^μ(z)≔ −∫log|z − s|dμ(s) is the logarithmic potential of μ [see Ref. 23 (Theorem I.3.3)].

Observe that due to the analyticity of the integrand in (1.1), Γ can be varied without changing the value of Z_N(t). Henceforth, we suppose that partition function (1.1) and orthogonal polynomials (1.10) are defined with $Γ \in T$ ⁠, where $T$ is the following class of contours:

Definition 2.2.

We shall denote by

T

the collection of all piecewise smooth contours that extend to infinity in both directions, each admiting a parameterization z(s),

s \in R

, for which there exists ɛ ∈ (0, π/6) and s₀ > 0 such that

\{\begin{cases} | a r g (z (s)) - π / 3 | \leq ε, & s \geq s_{0}, \\ | a r g (z (s)) - π | \leq ε, & s \leq - s_{0}, \end{cases})

(2.3)

where arg(z(s)) ∈ [0, 2π).

Despite the above flexibility, it is well understood in the theory of non-Hermitian orthogonal polynomials, starting with the works of Stahl^24–26 and Gonchar and Rakhmanov,²⁷ that one should use the contour whose equilibrium measure has support symmetric (with the S-property) in the external field ReV. We make this idea precise in the following definition:

Definition 2.3.

The support J_Γ has the S-property in the external field ReV if it consists of a finite number of open analytic arcs and their endpoints, and on each arc, it holds that

\frac{\partial}{\partial n_{+}} (2 U^{μ_{Γ}} + R e V) = \frac{\partial}{\partial n_{-}} (2 U^{μ_{Γ}} + R e V),

(2.4)

where

\frac{\partial}{\partial n_{+}}

and

\frac{\partial}{\partial n_{-}}

are the normal derivatives from the (+)- and (−)-sides of Γ. We shall say that a curve

Γ \in T

is an S-curve in the field ReV if J_Γ has the S-property in this field.

It is also understood that geometrically J_Γ is comprised of critical trajectories of a certain quadratic differential. Recall that if Q is a meromorphic function, a trajectory (respectively, orthogonal trajectory) of a quadratic differential −Q(z)dz² is a maximal regular arc on which

- Q (z (s)) {(z^{'} (s))}^{2} > 0 (respectively, - Q (z (s)) {(z^{'} (s))}^{2} < 0)

for any local uniformizing parameter. A trajectory is called critical if it is incident with a finite critical point [a zero or a simple pole of −Q(z)dz²] and it is called short if it is incident only with finite critical points. We designate the expression critical (orthogonal) graph of −Q(z)dz² for the totality of the critical (orthogonal) trajectories −Q(z)dz².

The following theorem is a specialization to V(z; t) of Ref. 22 (Theorems 2.3 and 2.4).

Theorem 2.1.

Let V(z; t) be given by (1.11).

There exists a contour $Γ_{t} \in T$ such that
$E_{V} (Γ_{t}) = \sup_{Γ \in T} E_{V} (Γ) .$
(2.5)
The support J_t of the equilibrium measure $μ_{t} ≔ μ_{Γ_{t}}$ has the S-property in the external field ReV(z; t), and the measure μ_t is uniquely determined by (2.4). Therefore, μ_t and its support J_t are the same for every Γ_t satisfying (2.5).
The function
$Q (z; t) = {(\frac{V^{'} (z; t)}{2} - \int \frac{d μ_{t} (s)}{z - s})}^{2}, z \in C \ J_{t},$
(2.6)
is a polynomial of degree 4.
The support J_t consists of some short critical trajectories of the quadratic differential −Q(z; t)dz², and the equation
$d μ_{t} (z) = - \frac{1}{π i} Q_{+}^{1 / 2} (z; t) d z, z \in J_{t},$
(2.7)
holds on each such critical trajectory, where $Q^{1 / 2} (z; t) = \frac{1}{2} z^{2} + O (z)$ as z → ∞ [in what follows, Q^1/2(z; t) will always stand for such a branch].

Much information on the structure of the critical graphs of a quadratic differential can be found in the excellent monographs.^28–30 Since deg Q = 4, J_t consists of one or two arcs, corresponding (respectively) to the cases where Q(z; t) has two simple zeros and one double zero and the case where it has four simple zeros. Away from J_t, one has freedom in choosing Γ_t. In particular, let

U (z; t) ≔ R e (2 \int_{e}^{z} Q^{1 / 2} (s; t) d s) = ℓ_{Γ_{t}} - R e (V (z; t)) - 2 U^{μ_{t}} (z),

(2.8)

where e ∈ J_t is any and the second equality follows from (2.6) [since the constant $ℓ_{Γ_{t}}$ in (2.2) is the same for both connected components of J_t and the integrand is purely imaginary on J_t, the choice of e is indeed not important]. Clearly, $U (z; t)$ is a subharmonic function (harmonic away from J_t), which is equal to zero J_t by (2.2). The trajectories of −Q(z; t)dz² emanating out of the endpoints of J_t belong to the set ${z : U (z; t) = 0}$ ⁠, and it follows from the variational condition (2.2) that $Γ_{t} \ J_{t} \subset {z : U (z; t) < 0}$ ⁠. However, within the region ${z : U (z; t) < 0}$ ⁠, the set Γ_t\J_t can be varied freely. The geometry of the set ${z : U (z; t) < 0}$ is described further below in Theorems 3.1 and 3.2.

III. STRUCTURE OF Γ_t

The partition of the phase space into one- and two-cut regions as well as the structure of Γ_t and its dependence on t has been heuristically described in Refs. 31 and 32. Even before these works, the phase diagram in Fig. 3 was heuristically described in Ref. 19. A mathematically rigorous description was provided in Ref. 3 but only in the one-cut region. Let us quickly recall the important notions from Ref. 3.

Denote by $C$ the critical graph of an auxiliary quadratic differential

- {(1 + 1 / s)}^{3} {d s}^{2}

(3.1)

[see Fig. 2(a)]. It was shown in Ref. 3 (Sec. 5) that $C$ consists of five critical trajectories emanating from −1 at the angles 2πk/5, k ∈ {0, 1, 2, 3, 4}, one of them being (−1, 0), other two forming a loop crossing the real line approximately at 0.635, and the last two approaching infinity along the imaginary axis without changing the half-plane (upper or lower). Given $C$ ⁠, define

Δ ≔ \{x : 2 x^{3} \in C\} .

Furthermore, put $Ω_{one−cut}$ to be the shaded region on Fig. 2(b) and set

\partial Ω_{one−cut} = Δ_{birth}^{b} \cup \{- 2^{- 1 / 3}\} \cup Δ_{split} \cup \{e^{π i / 3} 2^{- 1 / 3}\} \cup Δ_{birth}^{a},

where $Δ_{split}$ connects −2^−1/3 and e^πi/32^−1/3, $Δ_{birth}^{b}$ extends to infinity in the direction of the angle 7π/6, while $Δ_{birth}^{a}$ extends to infinity in the direction of the angle π/6. Let

t (x) ≔ (x^{3} - 1) / x

and set

\{\begin{cases} t_{cr} ≔ 3 \cdot 2^{- 2 / 3} = t (- 2^{- 1 / 3}), \\ O_{one−cut} ≔ t (Ω_{one−cut}), \\ C_{split} ≔ t (Δ_{split}), C_{birth}^{b} ≔ t (Δ_{birth}^{b}), C_{birth}^{a} ≔ t (Δ_{birth}^{a}), \\ S ≔ (t_{cr}, \infty), e^{2 π i / 3} S ≔ \{z : e^{- 2 π i / 3} z \in S\} \end{cases})

(3.2)

(see Fig. 3). The function t(x) is holomorphic in $Ω_{one−cut}$ with a non-vanishing derivative there. It maps $Ω_{one−cut}$ onto $O_{one−cut}$ in a one-to-one fashion. Hence, the inverse map x(t) exists and is holomorphic.

FIG. 2.

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Schematic representation of (a) the critical graph $C$ ⁠; (b) the set Δ (solid lines) and the domain $Ω_{one−cut}$ (shaded region).

FIG. 3.

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Domain $O_{one−cut}$ (shaded region); $\partial O_{one−cut}$ consisting of the open bounded arc $C_{split}$ ⁠, two open semi-unbounded arcs $C_{birth}^{a}$ and $C_{birth}^{b}$ ⁠, and two points $t_{cr}$ and $e^{2 π i / 3} t_{cr}$ ⁠; the semi-unbounded open horizontal rays S and e^2πi/3S (dashed lines).

Convention 1.

Below, we adopt the following convention: Γ(z₁, z₂) (respectively, Γ[z₁, z₂]) stands for the trajectory or orthogonal trajectory (respectively, the closure of) of the differential −Q(z; t)dz² connecting z₁ and z₂, oriented from z₁ to z₂, and $Γ (z, e^{i θ} \infty)$ [respectively, $Γ (e^{i θ} \infty, z)$ ] stands for the orthogonal trajectory ending at z, approaching infinity at the angle θ, and oriented away from z (respectively, oriented toward z).³³

The following theorem has been proven in Ref. 3 (Theorem 3.2), and it describes the geometry of Γ_t when $t \in {\bar{O}}_{one−cut}$ ⁠.

Theorem 3.1.

Let μ_t and Q(z; t) be as in Theorem 2.1, and let

J_{t} = s u p p (μ_{t})

. When

t \in {\bar{O}}_{one−cut}

, the polynomial Q(z; t) is of the form

Q (z; t) = \frac{1}{4} (z - a (t)) (z - b (t)) {(z - c (t))}^{2},

(3.3)

with a(t), b(t), and c(t) given by

\{\begin{cases} a (t) ≔ x (t) - i \sqrt{2} / \sqrt{x} (t), \\ b (t) ≔ x (t) + i \sqrt{2} / \sqrt{x} (t), \\ c (t) ≔ - x (t), \end{cases})

(3.4)

where

\sqrt{x} (t)

is the branch holomorphic in

O_{one−cut}

satisfying

\sqrt{x} (0) = e^{π i / 3}

. The set J_t consists of a single arc and the following:

If $t \in O_{one−cut}$ , then J_t = Γ[a, b] and an S-curve $Γ_{t} \in T$ can be chosen as
- $Γ (e^{π i} \infty, a) \cup J_{t} \cup Γ (b, e^{π i / 3} \infty)$ when t belongs to the connected component bounded by $S \cup C_{split} \cup e^{2 π i / 3} S$ [see Figs. 4(a)–4(e) ];
- $Γ (e^{π i} \infty, a) \cup J_{t} \cup Γ (b, c) \cup Γ (c, e^{π i / 3} \infty)$ when t ∈ S [see Fig. 4(f) ];
- $Γ (e^{π i} \infty, c) \cup Γ (c, a) \cup J_{t} \cup Γ (b, e^{π i / 3} \infty)$ when t ∈ e^2πi/3S;
- $Γ (e^{π i} \infty, a) \cup J_{t} \cup Γ (b, e^{- π i / 3} \infty) \cup Γ (e^{- π i / 3} \infty, c) \cup Γ (c, e^{π i / 3} \infty)$ when t belongs to the connected component bounded by $S \cup C_{birth}^{b}$ [see Fig. 4(g) ];
- $Γ (e^{π i} \infty, c) \cup Γ (c, e^{- π i / 3}) \cup Γ (e^{- π i / 3} \infty, a) \cup J_{t} \cup Γ (b, e^{π i / 3} \infty)$ when t belongs to the connected component bounded by $e^{2 π i / 3} S \cup C_{birth}^{a}$ .
If $t = t_{cr}$ (respectively, $t = e^{2 π i / 3} t_{cr}$ ), then J_t = Γ[a, b], c coincides with b (respectively, a), and an S-curve $Γ_{t} \in T$ can be chosen as in Case I(a) [see Fig. 5(a) ].
If $t \in C_{split}$ , then J_t = Γ[a, c] ∪ Γ[c, b] and an S-curve $Γ_{t} \in T$ can be chosen as in Case I(a) [see Fig. 5(b) ].
If $t \in C_{birth}^{b}$ (respectively, $t \in C_{birth}^{a}$ ), then J_t = Γ[a, b] and an S-curve $Γ_{t} \in T$ can be chosen as in Case I(d) [respectively, Case I(e)] [see Fig. 5(c) ].

FIG. 4.

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Schematic representation of the critical (solid) and critical orthogonal (dashed) graphs of −Q(z; t)dz² when $t \in O_{one−cut}$ ⁠. The bold curves represent the preferred S-curve Γ_t. Shaded region is the set ${U (z; t) < 0}$ ⁠.

FIG. 5.

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This is a continuation of Fig. 4 for the case $t \in \partial O_{one−cut}$ ⁠.

Now, let $O_{two−cut} ≔ C \ {\bar{O}}_{one−cut}$ ⁠. Then, the following theorem holds:

Theorem 3.2.

Let μ_t and Q(z; t) be as in Theorem 2.1,

J_{t} = s u p p (μ_{t})

. When

t \in O_{two−cut}

, the polynomial Q(z; t) is of the form

Q (z; t) = \frac{1}{4} (z - a_{1} (t)) (z - b_{1} (t)) (z - a_{2} (t)) (z - b_{2} (t)),

(3.5)

with a₁(t), b₁(t), a₂(t), and b₂(t) all distinct. The real and imaginary parts of a_i(t), b_i(t) are real analytic functions of Re(t) and Im(t) when

t \in O_{two−cut}

; however, at no point of

O_{two−cut}

any of the functions a_i(t), b_i(t) is analytic. The S-curve Γ_t can be chosen as

Γ (e^{π i} \infty, a_{1} (t)) \cup J_{t, 1} \cup Γ (b_{1} (t), e^{- π i / 3} \infty) \cup Γ (e^{- π i / 3} \infty, a_{2} (t)) \cup J_{t, 2} \cup Γ (b_{2} (t), e^{π i / 3} \infty),

where J_t = J_t,1 ∪ J_t,2 and

J_{t, i} ≔ Γ [a_{i} (t), b_{i} (t)]

, i ∈ {1, 2}; see Fig. 6 [this also explains how we choose the labeling of the zeros of Q(z; t) in the considered case]. Moreover, it holds that

a_{1} (t), b_{1} (t) \to a (t^{*}), b_{1} (t), a_{2} (t) \to c (t^{*}), and a_{2} (t), b_{2} (t) \to b (t^{*})

(3.6)

as t → t^* with

t^{*} \in C_{birth}^{a} \cup \{e^{2 π i / 3} t_{cr}\}

,

t^{*} \in C_{split}

, and

t^{*} \in C_{birth}^{b} \cup \{t_{cr}\}

, respectively.

FIG. 6.

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The schematic representation of the critical and critical orthogonal graphs of −Q(z; t)dz² when $t \in O_{two−cut}$ ⁠. The bold curves represent the preferred S-curve Γ_t. Shaded region is the set ${U (z; t) < 0}$ ⁠.

We prove Theorem 3.2 in Sec. V.

IV. MAIN RESULTS

In this section, we assume that $t \in O_{two−cut}$ and that Q(z; t) and J_t = J_t,1 ∪ J_t,2 are as in Theorem 3.2. We also put

E_{t} ≔ \{a_{1} (t), b_{1} (t), a_{2} (t), b_{2} (t)\}, J_{t, i} ° ≔ J_{t, i} \ E_{t}, and J_{t} ° ≔ J_{t, 1} ° \cup J_{t, 2} ° .

Convention 2.

When it comes to the definition of the S-contour Γ_t, it will be more practical for us to change the choice of Γ_t made in Theorem 3.2. Rather than including unbounded trajectories $Γ (b_{1} (t), e^{- π i / 3} \infty)$ and $Γ (e^{- π i / 3} \infty, a_{2} (t))$ , we shall replace them by a smooth Jordan arc, say, I_t, connecting b₁(t) and a₂(t) such that I_t° ≔ I_t\E_t lies entirely within the region ${U (z; t) < 0}$ while there exist $s_{1} (t) \in Γ (b_{1} (t), e^{- π i / 3} \infty)$ and $s_{2} (t) \in Γ (e^{- π i / 3} \infty, a_{2} (t))$ for which $Γ (b_{1} (t), s_{1} (t)) \cup Γ (s_{2} (t), a_{2} (t)) \subset I_{t} °$ (see Fig. 14 ).

To describe the asymptotics of the orthogonal polynomials themselves, we need to construct the Szegő function of e^V(z;t).

Proposition 4.1.

Let constants ς(t) and d_i(t) be given by

ς (t) ≔ \frac{2 t}{3 d_{0} (t)} and d_{i} (t) ≔ \int_{I_{t}} \frac{s^{i} d s}{Q^{1 / 2} (s; t)},

(4.1)

where, as usual, we use the branch

Q^{1 / 2} (z; t) = \frac{1}{2} z^{2} + O (z)

as z → ∞. Then, the Szegő function

D (z; t) ≔ \exp \{\frac{1}{2} V (z; t) + \frac{1}{3} (z + \int_{I_{t}} \frac{3 ς (t)}{s - z} \frac{d s}{Q^{1 / 2} (s; t)}) Q^{1 / 2} (z; t)\}

(4.2)

is holomorphic and non-vanishing in

\bar{C} \ Γ [a_{1} (t), b_{2} (t)]

with continuous traces on J_t° ∪ I_t° that satisfy

\{\begin{cases} D_{+} (s; t) D_{-} (s; t) = e^{V (s; t)}, & s \in J_{t} °, \\ D_{+} (s; t) = D_{-} (s; t) e^{2 π i ς (t)}, & s \in I_{t} ° . \end{cases})

(4.3)

Denote by μ₁(t) ≔ ∫sdμ_t(s) the first moment of the equilibrium measure μ_t. Then,

D (z; t) = \exp \{\frac{1}{3} - \frac{(ς d_{1}) (t)}{2}\} (1 + \frac{t^{2} + μ_{1} (t) - 3 (ς d_{2}) (t) / 2}{3 z} + O (\frac{1}{z^{2}}))

(4.4)

as z → ∞. In particular, it holds that

D (\infty; t) = \exp \{\frac{1 - t (d_{1} / d_{0}) (t)}{3}\}

.

We shall also denote by $D (z; t) ≔ D (z; t) / D (\infty; t)$ the normalized Szegő function. We prove Proposition 4.1 in Sec. VI A.

To describe the geometric growth of the orthogonal polynomials, let us define

Q (z; t) ≔ \int_{b_{2} (t)}^{z} Q^{1 / 2} (s; t) d s, z \in C \ Γ (e^{π i} \infty, b_{2} (t)]

(4.5)

(where we deviate slightly from Convention 1 and denote by Γ[s₁, s₂] a subarc of Γ_t connecting s₁ to s₂; for example, Γ(e^πi∞, a₂] includes I_t rather than the short trajectories connecting b₁ to a₂). Observe that $U (z; t) = 2 R e (Q (z; t))$ as defined in (2.8). This function has the following properties:

Proposition 4.2.

Let constants τ(t), ω(t) be given by

τ (t) ≔ - \frac{1}{π i} \int_{I_{t}} Q^{1 / 2} (s; t) d s and ω (t) ≔ - \frac{1}{π i} \int_{J_{t, 1}} Q_{+}^{1 / 2} (s; t) d s .

(4.6)

These constants are necessarily real [in fact, ω(t) = μ_t(J_t,1); see (2.7)]. The function

e^{Q (z; t)}

is holomorphic in

C \ Γ [a_{1} (t), b_{2} (t)]

⁠, and there exists a constant ℓ_*(t) such that

\exp \{\frac{V (z; t) - ℓ_{*} (t)}{2} + Q (z; t)\} = z + O (1) as z \to \infty .

(4.7)

Moreover,

Q (z; t)

possesses continuous traces on J_t° ∪ I_t° that are purely imaginary on J_t° and satisfy

\{\begin{cases} e^{Q_{+} (s; t) {+ Q}_{-} (s; t)} = 1, & s \in J_{t, 2} °, \\ e^{Q_{+} (s; t) {+ Q}_{-} (s; t)} = e^{2 π i τ (t)}, & s \in J_{t, 1} °, \\ e^{Q_{+} (s; t)} = e^{Q_{-} (s; t) - 2 π i ω (t)}, & s \in I_{t} ° . \end{cases})

(4.8)

We prove Proposition 4.2 in Sec. VI B. Observe that it follows from Theorem 3.2 that $| e^{Q (z; t)} |$ is less than 1 when $U (z; t) < 0$ (the shaded areas of Fig. 6), is equal to 1 on the critical trajectories (black curves), and otherwise is greater than 1. The constant ℓ_*(t) is such that $R e (ℓ_{*} (t)) = ℓ_{Γ_{t}}$ [see (2.2)]. Moreover, the following identity holds:

Proposition 4.3.

Set

B (t) ≔ - (\int_{J_{t, 1}} \frac{d s}{Q_{+}^{1 / 2} (s; t)}) / (\int_{I_{t}} \frac{d s}{Q^{1 / 2} (s; t)}) .

(4.9)

Let

θ (u; B (t))

be the Riemann theta function associated with

B (t)

, i.e.,

θ (u; B (t)) = \sum_{k \in Z} \exp \{π i B (t) k^{2} + 2 π i u k\}, u \in C .

(4.10)

Then, it holds that

e^{ℓ_{*} (t)} = {(4 D (\infty; t) \frac{e^{π i τ (t) (ς (t) + ω (t) + B (t) τ (t))}}{b_{1} (t) - a_{1} (t) + b_{2} (t) - a_{2} (t)} \frac{θ (ς (t) + ω (t) + B (t) τ (t))}{θ (0)})}^{2} .

(4.11)

We prove this proposition in Sec. VII F. We also explain right after the statement of Proposition 7.1 further below that b₁(t) − a₁(t) + b₂(t) − a₂(t) ≠ 0. Another auxiliary function we need is given by

A (z; t) ≔ \frac{1}{2} ({(\frac{z - b_{2} (t)}{z - a_{2} (t)} \frac{z - b_{1} (t)}{z - a_{1} (t)})}^{1 / 4} + {(\frac{z - b_{2} (t)}{z - a_{2} (t)} \frac{z - b_{1} (t)}{z - a_{1} (t)})}^{- 1 / 4})

(4.12)

for $z \in \bar{C} \ J_{t}$ ⁠, where the branches are chosen so that the summands are holomorphic in $\bar{C} \ J_{t}$ and have value 1 at infinity. As explained in Sec. VII B, this function can be analytically continued through each side of J_t° and is non-vanishing in the domain of the definition.

Given a sequence ${N_{n}}_{n \in N}$ ⁠, we define further below in (7.23) functions Θ_n(z; t), which are ratios of Riemann theta functions (4.10) with various arguments. To shorten the presentation of the main results, we only discuss main features of the functions Θ_n(z; t) and defer the detailed construction and description of further properties to Sec. VII.

Proposition 4.4.

Functions Θ_n(z; t) are holomorphic in

\bar{C} \ Γ [a_{1} (t), b_{2} (t)]

with at most one zero there. They have continuous traces on I_t° that satisfy

Θ_{n +} (s; t) = Θ_{n -} (s; t) e^{2 π i (n ω (t) + (n - N_{n}) ς (t))} .

Assume that there exists a constant N_* such that |n − N_n| ≤ N_* for all

n \in N

. Then, for any δ > 0, there exists a constant C(t, δ, N_*) such that

| A (z; t) Θ_{n} (z; t) | \leq C (t, δ, N_{*}), z \in \bar{C} \ ⋃_{e \in E_{t}} \{| z - e | < δ\},

that is, including the traces on Γ[a₁(t), b₂(t)]. Given ɛ > 0, let

N (t, ε)

be a subsequence of indices n such that Θ_n(z; t) is non-vanishing in {|z| ≥ 1/ɛ}. Then, there exists a constant c(t, ɛ) > 0 such that

| Θ_{n} (\infty; t) | \geq c (t, ε), n \in N (t, ε) .

As in the case of the Szegő functions $D (z; t)$ ⁠, it would be convenient for us to renormalize Θ_n(z; t) at infinity. Thus, we set ϑ_n(z; t) ≔ Θ_n(z; t)/Θ_n(∞; t). Observe that $ϑ_{n} (z; t) D^{N_{n} - n} (z; t) e^{n Q (z; t)}$ are functions holomorphic in $C \ J_{t}$ ⁠.

Proposition 4.4 has substance only if the sets $N (t, ε)$ have infinite cardinality. Recall $B (t)$ from (4.9). It follows from the general theory of Riemann surfaces (see Sec. VII A) that $I m (B (t)) > 0$ ⁠. In particular, any $s \in C$ can be uniquely written as $x + B (t) y$ for some $x, y \in R$ ⁠.

Proposition 4.5.

Write $ς (t) = : x (t) + B (t) y (t)$ , $x (t), y (t) \in R$ . It holds that

if $x (t), y (t) \in Z$ , then the functions Θ_n(z; t) do not depend on the choice of N_n and at least one of the integers n, n + 1 belongs to $N (t, ε)$ ;³⁴
otherwise, for any natural number N_* > 0, there exists a choice of N_n such that |n − N_n| ≤ N_* and $N (t, ε) = N$ for all ɛ small enough.

On the other hand, if the sequence ${N_{n}}_{n \in N}$ is fixed, we can claim the following proposition:

Proposition 4.6.

Let

{N_{n}}_{n \in N}

be a sequence such that |n − N_n| ≤ N_* for some N_* ≥ 0. The sequence

N (t, ε)

is infinite for all ɛ small enough unless there exist integers d > 0, k, i₁, i₂, m₁, m₂ such that

ς (t) = (i_{1} + B (t) i_{2}) / d, ω (t) d = (k - 1) i_{1} + m_{1} d, and τ (t) d = (k - 1) i_{2} + m_{2} d,

(4.13)

where at least one of the fractions i₁/d, i₂/d is irreducible, d is even, i₁ and i₂ are odd, and the sequence {N_n} is such that all but finitely many numbers nk − N_n are divisible by d/2 but not by d. Moreover, the following special cases take place:

If there exists an infinite subsequence {n_l} such that $N_{n_{l} + 1} - N_{n_{l}} \in {0, 1}$ (in particular, this happens when N_n = n, in which case the differences are always equal to 1), then at least one of the integers n_l, n_l + 1 belongs to $N (t, ε)$ .
If one of the triples ω(t), x(t), 1 or τ(t), y(t), 1 is rationally independent, then at least one of the integers n, n + 1 belongs to $N (t, ε)$ .

We prove Propositions 4.4–4.6 in Secs. VII C–VII E, respectively.

Theorem 4.1.

Let

t \in O_{two−cut}

and

{N_{n}}_{n = 1}^{\infty}

be a sequence such that |n − N_n| ≤ N_* for some N_* fixed. Let P_n(z; t, N) be defined by () and () and

ψ_{n} (z; t) ≔ P_{n} (z; t, N_{n}) e^{- n (V (z; t) - ℓ_{*} (t)) / 2} .

Given ɛ > 0, let

N (t, ε)

be as in Proposition 4.4. Then, for all

n \in N (t, ε)

large enough, it holds that³⁵

ψ_{n} (z; t) = ((A ϑ_{n} D^{N_{n} - n}) (z; t) {+ O}_{ε} (n^{- 1})) e^{n Q (z; t)}

(4.14)

locally uniformly in

C \ J_{t}

; moreover,

ψ_{n} (s; t) = {(A ϑ_{n} D^{N_{n} - n})}_{+} (s; t) e^{{n Q}_{+} (s; t)} + {(A ϑ_{n} D^{N_{n} - n})}_{-} (s; t) e^{{n Q}_{-} (s; t)} {+ O}_{ε} (n^{- 1})

(4.15)

locally uniformly on J_t°.

Recall that each ϑ_n(z; t) might have a single zero in $C \ J_{t}$ ⁠. If these zeros accumulate at a point z_* along a subsequence of $N (t, ε)$ ⁠, then the polynomials P_n(z; t, N_n) will have a single zero approaching z_* along this subsequence by (4.14) and Rouche’s theorem. With this exception, it also follows from (4.14) that P_n(z; t, N_n) are eventually zero free on compact subsets $C \ J_{t}$ ⁠. The main part of the Proof of Theorem 4.1 is carried out in Sec. VIII with auxiliary details relegated to Secs. VI and VII.

The functions ϑ_n(z; t) were defined as pull-backs from one of the sheets (a copy of $\bar{C} \ J_{t}$ ⁠) of certain ratios of Riemann theta functions defined on the (two-sheeted) Riemann surface of Q^1/2(z; t). To describe asymptotics of the recurrence coefficients, we shall need the pull-backs from the other sheet as well, which, for the purpose of the next theorem, we denote by $ϑ_{n}^{*} (z; t)$ ⁠. To complicate matters further, our analysis requires another related family of ratios of theta functions, which, in the next theorem, we will denote by ${\tilde{ϑ}}_{n} (z; t)$ [see (7.24) for the rigorous definition of these functions]. The Proof of Proposition 4.4, in fact, shows that $ϑ_{n}^{*} (\infty; t) {= O}_{ε} (1)$ and ${\tilde{ϑ}}_{n} (\infty; t) {= O}_{ε} (1)$ for $n \in N (t, ε)$ ⁠.

Theorem 4.2.

Let

S_{k} (t) ≔ \sum_{i = 1}^{2} (b_{i}^{k} (t) - a_{i}^{k} (t))

for k ∈ {1, 2}. In the setting of Theorem 4.1, it holds for all

n \in N (t, ε)

large enough that

\{\begin{cases} h_{n} (t, N_{n}) = \frac{π}{2} e^{- n ℓ_{*} (t)} S_{1} {(t) D}^{2 (n - N_{n})} (\infty; t) ϑ_{n}^{*} (\infty; t) {+ O}_{ε} (n^{- 1}), \\ γ_{n}^{2} (t, N_{n}) = \frac{1}{16} S_{1}^{2} (t) ϑ_{n}^{*} (\infty; t) {\tilde{ϑ}}_{n} (\infty; t) {+ O}_{ε} (n^{- 1}), \end{cases})

(4.16)

where

D (\infty; t)

is as in Proposition 4.1, ℓ_*(t) as in (4.7), and

O_{ε} (\cdot)

is interpreted in the same way as in Theorem 4.1. Furthermore, it holds that

β_{n} (t, N_{n}) = \frac{ϑ_{n}^{*} (\infty; t) (\frac{S_{2} (t)}{2 S_{1} (t)} + \frac{d}{d z} {((\log ϑ_{n} (1 / z; t) + \log ϑ_{n}^{*} (1 / z; t))|}_{z = 0}) {+ O}_{ε} (n^{- 1})}{ϑ_{n}^{*} (\infty; t) {+ O}_{ε} (n^{- 1})},

(4.17)

where we agree that

ϑ_{n}^{*} (\infty; t) {(\frac{d}{d z} (\log ϑ_{n}^{*} (1 / z; t))|}_{z = 0} = {(\frac{d}{d z} ϑ_{n}^{*} (1 / z; t)|}_{z = 0}

even when

ϑ_{n}^{*} (\infty; t) = 0

.

Of course, if there exists a subsequence of $N (t, ε)$ along which the values $ϑ_{n}^{*} (\infty; t)$ are separated away from zero, then formula (4.17) can be further simplified. However, such a subsequence might not exist [in particular, one can deduce from (7.7) and (7.20) that it does not exist when N_n = n and ω(t) = τ(t) = 1/2]. Note that formulas (4.16) are not immediately reflective of the equality in (1.14). However, as we point out further in (9.12), the leading term of the asymptotics of $γ_{n}^{2} (t, N_{n})$ can be rewritten to make this connection more transparent. Theorem 4.2 is proven in Sec. IX, but its proof heavily relies on the material of Sec. VIII.

V. S-CURVES

In this section, we prove Theorem 3.2. We do it in several steps. In Sec. V A, we gather results about quadratic differentials that will be important to us throughout the proof. In Sec. V B, we show the validity of formula (3.5); that is, we prove that we are indeed in the two-cut case when $t \in O_{two−cut}$ ⁠. In Sec. V C, we show that the critical and critical orthogonal graphs of

ϖ_{t} (z) ≔ - Q (z; t) d z^{2}

do look like as depicted in Fig. 6. In Sec. V D, we describe the dependence of the zeros of Q(z; t) on t by showing that the variational condition (2.2) and the S-property (2.6) yield that the zeros satisfy a certain system of real equations with non-zero Jacobian [see (5.17) and (5.18)] and that this system is, in fact, uniquely solved by them. Finally, in Sec. V E, we establish the limits in (3.6).

A. On quadratic differentials

To start, let us also recall the following important result, known as Teichmüller’s lemma [see Ref. 30 (Theorem 14.1)]. Let P be a geodesic polygon of a quadratic differential, that is, a Jordan curve in $\bar{C}$ that consists of a finite number of trajectories and orthogonal trajectories of this differential. Then, it holds that

\sum_{z \in P} (1 - θ (z) \frac{2 + o r d (z)}{2 π}) = 2 + \sum_{z \in i n t (P)} o r d (z),

(5.1)

where ord(z) is the order of z with respect to the considered differential and θ(z) ∈ [0, 2π], z ∈ P, is the interior angle of P at z. Both sums in (5.1) are finite since only critical points of the differential have a non-zero contribution.

Let us briefly recall the main properties of the differential ϖ_t(z). The only critical points of ϖ_t(z) are the zeros of Q(z; t) and the point at infinity. Regular points have order 0, the order of a zero of Q(z; t) is equal to its multiplicity, and infinity is a critical point of order −8. Through each regular point passes exactly one trajectory and one orthogonal trajectory of ϖ_t(z), which are orthogonal to each other at the point. Two distinct (orthogonal) trajectories meet only at critical points [see Ref. 30 (Theorem 5.5)]. As Q(z; t) is a polynomial, no finite union of (orthogonal) trajectories can form a closed Jordan curve, while a trajectory and an orthogonal trajectory can intersect at most once [see Ref. 29 (Lemma 8.3)]. Furthermore, (orthogonal) trajectories of ϖ_t(z) cannot be recurrent (dense in two-dimensional regions) [see Ref. 28 (Theorem 3.6)]. From each critical point of order m > 0, there emanate m + 2 critical trajectories whose consecutive tangent lines at the critical point form an angle 2π/(m + 2). Furthermore, since infinity is a pole of order 8, the critical trajectories can approach infinity only in six distinguished directions, namely, asymptotically to the lines L_−π/6, L_π/6, and L_π/2, where L_θ = {z: z = re^iθ, r ∈ (−∞, ∞)}. In fact, there exists a neighborhood of infinity such that any trajectory entering it necessarily tends to infinity [see Ref. 30 (Theorem 7.4)]. This discussion also applies to orthogonal trajectories. In particular, they can approach infinity asymptotically to the lines L₀, L_π/3, and L_2π/3 only.

Denote by $G$ the critical graph of ϖ_t(z), that is, the totality of all the critical trajectories of ϖ_t(z). Then [see Ref. 28 (Theorem 3.5)], the complement of $G$ can be written as a disjoint union of either half-plane or strip domains. Recall that a half-plane (or end) domain is swept by trajectories unbounded in both directions that approach infinity along consecutive critical directions. Its boundary is connected and consists of a union of two unbounded critical trajectories and a finite number (possibly zero) of short trajectories of ϖ_t(z). The map $z \mapsto \int^{z} \sqrt{- ϖ_{t}}$ maps end domains conformally onto half planes ${z \in C | R e (z) > c}$ for some $c \in R$ that depends on the domain and extends continuously to the boundary. Similarly, a strip domain is again swept by trajectories unbounded in both directions, but its boundary consists of two disjoint ϖ_t(z)-paths, each of which is comprised of two unbounded critical trajectories and a finite number (possibly zero) of short trajectories. The map $z \mapsto \int^{z} \sqrt{- ϖ_{t}}$ maps strip domains conformally onto vertical strips ${w \in C | c_{1} < R e (w) < c_{2}}$ for some $c_{1}, c_{2} \in R$ depending on the domain and extends continuously to their boundaries. The number c₂ − c₁ is known as the width of a strip domain and can be calculated in terms of ϖ_t(z) as

|R e \int_{p}^{q} \sqrt{- ϖ_{t}}|,

(5.2)

where p, q belong to different components of the boundary of the domain.

B. Proof of (3.5)

Assume to the contrary that for a given $t \in O_{two−cut}$ ⁠, we are in one-cut case. That is, there exists a choice of a, b, and c such that the polynomial Q(z; t) from (2.6) has the form (3.3). It follows from (2.6) in conjunction with (1.11) that

Q (z; t) = {(\frac{- z^{2} + t}{2} - \frac{1}{z} + O (z^{- 2}))}^{2} = \frac{{(z^{2} - t)}^{2}}{4} + z + C

(5.3)

for some constant C. Then, by equating the coefficients in (3.3) and (5.3), we obtain a system of equations

\{\begin{aligned} a + b + 2 c = 0, \\ a b + c^{2} + 2 (a + b) c = - 2 t, \\ 2 a b c + (a + b) c^{2} = - 4 . \end{aligned})

(5.4)

Setting x ≔ (a + b)/2 and eliminating the product ab from the second and third relations in (5.4) yield

x^{3} - t x - 1 = 0,

(5.5)

which is exactly the equation appearing before (3.2). Given any solution of (5.5), say, x(t), then a(t), b(t), and c(t) are necessarily expressed via (3.4). Theorem 2.1 and the variational condition (2.2) imply that there must exist a contour $Γ_{t} \in T$ [this class of contours was defined right after (2.3)] such that

U (z; t) \leq 0 for all z \in Γ_{t}

(5.6)

[see (2.8)]. In what follows, we shall show that no such contour exists in $T$ for any of the three possible choices of x(t) solving (5.5) when $t \in O_{two−cut}$ and Q(z; t) is given by (3.3) and (3.4).

In accordance with the above strategy, observe that the solutions of (5.5) can be written as

x_{k} (t) = u_{k} (t) + \frac{t}{3 u_{k} (t)}, u_{k} (t) ≔ {(\frac{1}{2} - \sqrt{\frac{1}{4} - \frac{t^{3}}{27}})}^{1 / 3} e^{2 k π i / 3},

(5.7)

k ∈ {0, 1, 2}, with all branches being principal. It can be readily verified that x₁(t) is analytic in $C \ (\bar{e^{2 π i / 3} S}, \bar{S})$ (here, $\bar{\cdot}$ means topological closure) [see (3.2) for the definition of the ray S] and

\{\begin{cases} x_{0} (t) = e^{4 π i / 3} x_{1} (t e^{- 2 π i / 3}), \\ x_{1} (t) = e^{4 π i / 3} \bar{x_{1} (\bar{t} e^{2 π i / 3})}, \\ x_{2} (t) = \bar{x_{1} (\bar{t})} . \end{cases})

(5.8)

Furthermore, noting that the function x(t), defined after (3.2), maps $Ω_{one−cut}$ onto $O_{one−cut}$ ⁠, it can be easily checked that x(t) is evaluated as shown on Fig. 7, where the dashed lines are the chosen branch cuts of x₁(t). In particular, x(t) can be analytically continued across $C_{split}$ ⁠, $C_{birth}^{a}$ ⁠, and $C_{birth}^{b}$ [see (3.2) and Fig. 7]. In what follows, we consider what happens in the case of each of these continuations.

FIG. 7.

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Determination of x(t).

Continue x(t) into $O_{two−cut}$ by either x₂(t) or x₀(t), that is, analytically across either $C_{birth}^{b}$ or $C_{birth}^{a}$ (see Fig. 7). The first and the last symmetries in (5.8) then yield that ϖ_t(z) is either equal to

\bar{ϖ_{\bar{t}} (\bar{z})} or ϖ_{t e^{- 2 π i / 3}} (z e^{- 4 π i / 3}) .

Since the set $O_{two−cut}$ is symmetric with respect to the line L_π/3, its rotation by −2π/3 is equal to its reflection across the real axis. Thus, the critical graph ϖ_t(z) when $t \in O_{two−cut}$ and x(t) is continued by either x₂(t) or x₀(t) being equal to the reflection across the real axis or the rotation by 4π/3 of the critical graph of $ϖ_{t_{*}} (z)$ for some t_* such that ${\bar{t}}_{*} \in O_{two−cut}$ ⁠. These graphs were studied in Ref. 3 (Theorems 3.2 and 3.4) and determined to have the structure as depicted on Figs. 4(a)–4(c) (or the reflection of these three panels across the line L_2π/3). A direct examination shows that none of these critical graphs form a curve in $T$ for which (5.6) holds (such a curve must belong to the closure of the gray regions on Fig. 4).

Suppose now that we continue x(t) by x₁(t), that is, analytically across $C_{split}$ ⁠. For such a choice of x(t), the critical graph of ϖ_t(z) was studied in Ref. 21 when t ∈ L_π/3 (in which case the critical graph is symmetric with respect to L_2π/3). In particular, it was shown that x(t) ∈ L_2π/3, no union of critical trajectories join a and b, and no critical trajectory of ϖ_t(z) crosses the line L_2π/3 when $t \in L_{π / 3} \cap O_{two−cut}$ [see Ref. 21 (Lemma 3.2)]. Since critical trajectories cannot intersect, can approach infinity asymptotically to the lines L_π/6, L_π/2, and L_−π/6 only, and must obey Teichmüller’s lemma (5.1), the critical graph of ϖ_t(z) must be as on Fig. 8. Clearly, this critical graph does not yield a curve in $T$ for which (5.6) holds. Thus, to complete the proof, we need to argue that the structure of the critical trajectories of ϖ_t(z) remains the same for all $t \in O_{two−cut}$ in the considered case. Observe that it is enough to show that all the trajectories out of b approach infinity.

FIG. 8.

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The critical graph of ϖ_t(z) when x(t) is analytically continued across $C_{split}$ ⁠.

Recall that the trajectories emanating out of b are part of the level set ${U (z; t) = 0}$ [see (2.8)]. When $t \in L_{π / 3} \cap O_{two−cut}$ ⁠, these trajectories approach infinity at the angles −π/6, π/6, and π/2 (see Fig. 8). Since the values of $U (z; t)$ analytically depend on t, the same must be true in a neighborhood of each such t. This will remain so until one of the trajectories hits a critical point different from the one at infinity. As can be seen from Fig. 8, this critical point must necessarily be c = −x [see (3.4)]. That is, as long as $U (- x; t) \neq 0$ ⁠, the trajectories out of b will asymptotically behave as on Fig. 8. When x(t) is continued into $O_{two−cut}$ by x₁(t), its values lie within the gray region on Fig. 9(b) [see also Fig. 2(b)]. The values 2x³(t) lie within the gray region on Fig. 9(a) [see also Fig. 2(a)]. It was verified in Ref. 3 (Sec. 5.3) that

U (- x; t) = R e (\frac{2}{3} \int_{- 1}^{2 x^{3}} {(1 + \frac{1}{s})}^{3 / 2} d s),

where the path of integration lies within the shaded domain on Fig. 9(a). Hence, $U (- x; t) = 0$ if and only if 2x³ belongs to a trajectory of −(1 + 1/s)³ds² emanating from −1. These trajectories are drawn on Figs. 2 and 9(a) (black lines). Thus, $U (- x; t) \neq 0$ in the considered case as claimed.

FIG. 9.

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Shaded regions represent the domains within which $2 x_{1}^{3} (t)$ [panel (a)] and x₁(t) [panel (b)] change when $t \in O_{two−cut}$ ⁠.

C. Critical graph of ϖ_t(z)

Let, as usual, Q(z; t) be the polynomial guaranteed by Theorem 2.1. According to what precedes, it has the form (3.5) when $t \in O_{two−cut}$ ⁠. Recall the properties of the differential ϖ_t(z) = −Q(z; t)dz² described at the beginning of Sec. V A. In particular, it has four critical points of order 1, which, for a moment, we label as z₁(t), z₂(t), z₃(t), z₄(t) [these are the zeros of Q(z; t)], a critical point of order −8 at infinity, and no other critical points. It follows from Theorem 2.1(4) and (2.2) with (2.8) that

R e (\int_{Γ_{t} [z_{i} (t), z_{j} (t)]} Q_{+}^{1 / 2} (z; t) d z) = 0,

(5.9)

where Γ_t[z_i(t), z_j(t)] is the subarc of Γ_t with endpoints z_i(t), z_j(t) and $Q_{+}^{1 / 2} (z; t)$ is the trace of Q^1/2(z; t) on the positive side of Γ_t. Equation (5.9) implies the existence of three short critical trajectories of ϖ_t(z). Indeed, if all three critical trajectories out of a zero z_i(t) approach infinity, then z_i(t) must belong to a boundary of at least one strip domain. Let z_j(t) be a different zero of Q(z; t) belonging to the other component of the boundary of this strip domain. Then, it follows from (5.2) and (5.9) that the width of this strip domain is 0, which is impossible. Thus, each zero of Q(z; t) must be coincident with at least one short trajectory. Therefore, either there is a zero, say, z₄(t), connected by short trajectories to the remaining three zeros or there are at least two short trajectories connecting two pairs of zeros. In the latter case, label these zeros by a₁(t), b₁(t) and a₂(t), b₂(t). If the other two trajectories out of both a₁(t) and b₁(t) approach infinity, one of these zeros again must belong to the boundary of a strip domain with either a₂(t) or b₂(t) belonging to the other component of the boundary. As before, (5.9) yields that the width of this strip domain is 0, which, again, is impossible. Thus, in this case, there also exists a third short critical trajectory. Then, we choose a labeling of the zeros so that b₁(t) and a₂(t) are connected by this trajectory.

Since short critical trajectories cannot form closed curves, there cannot be any more of them. That is, the remaining critical trajectories are unbounded. Consider the two unbounded critical trajectories out of z₁(t) in the case where short ones form a threefold [see Fig. 10(c)]. Since critical trajectories cannot intersect and the remaining zeros are connected to z₁(t) by short critical trajectories, the unbounded critical trajectories out of z₁(t) delimit a half-plane domain and, in particular, must approach infinity along consecutive critical directions [those are given by the angles (2k + 1)π/6, k ∈ {0, …, 5}; see Sec. V A]. Clearly, the same is true for the unbounded critical trajectories out of z₂(t) and z₃(t) as well as for the unbounded critical trajectories out of a₁(t), b₂(t), and the union of the unbounded critical trajectory out of b₁(t), the short critical trajectory connecting b₁(t) to a₂(t), and the unbounded critical trajectory out of a₂(t) in the case where short critical trajectories form a Jordan arc (see Fig. 10).

FIG. 10.

$FIG. 10. Geometries of the critical graph of ϖt(z). Shaded regions represent the open set {U(z;t)<0}, the white regions represent the open {U(z;t)>0}, and the red dashed arcs form Γt\Jt.$

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Geometries of the critical graph of ϖ_t(z). Shaded regions represent the open set ${U (z; t) < 0}$ ⁠, the white regions represent the open ${U (z; t) > 0}$ ⁠, and the red dashed arcs form Γ_t\J_t.

Now, let $U (z; t)$ be given by (2.8). Clearly, $U (z; t)$ is a subharmonic function that is equal to zero on J_t [see (2.2)]. Since $U (z; t)$ must have the same sign on both sides of each subarc of J_t by (2.4), it follows from the maximum principle for subharmonic functions that it is positive there. Furthermore, since trajectories of ϖ_t(z) cannot form a closed Jordan curve, all the connected components of the open set ${U (z; t) < 0}$ must necessarily extend to infinity. Since Re(V(z; t)) is the dominant term of $U (z; t)$ around infinity [see (2.8)], for any δ > 0, there exists R > 0 sufficiently large so that

\{\begin{cases} (S_{π / 3, δ} \cup S_{π, δ} \cup S_{- π / 3, δ}) \cap {| z | > R} & \subset {U (z; t) < 0}, \\ (S_{0, δ} \cup S_{2 π / 3, δ} \cup S_{- 2 π / 3, δ}) \cap {| z | > R} & \subset {U (z; t) > 0}, \end{cases})

(5.10)

where S_θ,δ ≔ {|arg(z) − θ| < π/6 − δ}. Altogether, the critical graph of ϖ_t(z) must look like either on Fig. 6 or on Fig. 10.

It remains to show that ϖ_t(z) cannot have the critical graph as on any of the panels of Fig. 10. To this end, recall that the contour Γ_t must contain J_t and two unbounded arcs extending to infinity in the directions π/3 and π (red dashed unbounded arcs on Fig. 10). Let Γ_* be obtained from Γ_t by dropping the short trajectory that is a part of J_t and whose removal keeps Γ_* connected (this can be done for any of the panels on Fig. 10). Observe that Γ_* also belongs to $T$ ⁠. Let μ_* be the weighted equilibrium distribution on Γ_* as defined in Definition 2.1. Since Γ_* ⊂ Γ_t, it holds that $μ_{*} \in M (Γ_{t})$ ⁠. Moreover, since μ_* ≠ μ_t, $E_{V} (Γ_{*}) = E_{V} (μ_{*}) > E_{V} (μ_{t}) {= E}_{V} (Γ_{t})$ [see (2.1)]. However, the last inequality clearly contradicts (2.5).

We have shown that the critical graph of ϖ_t(z) must look like on Fig. 6. As the critical orthogonal and critical trajectories cannot intersect, the structure of the critical orthogonal graph is uniquely determined by structure of the critical graph. Now, we can completely fix the labeling of the zeros of Q(z; t) by given the label a₁(t) to one that is incident with the orthogonal critical trajectory extending to infinity asymptotically to the ray arg(z) = π.

D. Dependence on t

We start with some general considerations. Let f(z) and g(z) be analytic functions of z = x + iy. Consider a determinant of the form

D = |\begin{matrix} \partial_{x} R e (f) & \partial_{y} R e (f) & * \\ \partial_{x} I m (f) & \partial_{y} I m (f) & * \\ \partial_{x} R e (g) & \partial_{y} R e (g) & * \end{matrix}|,

where the entries of the third column are not important for the forthcoming computation. Due to Cauchy–Riemann relations, it holds that f′ = ∂_x Re(f) + i∂_x Im(f) = ∂_y Im(f) − i∂_y Re(f). Therefore,

D = |\begin{matrix} R e (f^{'}) & - I m (f^{'}) & * \\ I m (f^{'}) & R e (f^{'}) & * \\ R e (g^{'}) & - I m (g^{'}) & * \end{matrix}| = |\begin{matrix} f^{'} & i f^{'} & * \\ I m (f^{'}) & R e (f^{'}) & * \\ R e (g^{'}) & - I m (g^{'}) & * \end{matrix}| = \frac{i}{2} |\begin{matrix} f^{'} & i f^{'} & * \\ \bar{f^{'}} & - i \bar{f^{'}} & * \\ R e (g^{'}) & - I m (g^{'}) & * \end{matrix}|

by adding the second row times i to the first one and then multiplying the second row by −2i and adding the first row to it. It further holds that

D = \frac{i}{2} |\begin{matrix} 2 f^{'} & i f^{'} & * \\ 0 & - i \bar{f^{'}} & * \\ g^{'} & - I m (g^{'}) & * \end{matrix}| = \frac{i}{2} |\begin{matrix} 2 f^{'} & 0 & * \\ 0 & - i \bar{f^{'}} & * \\ g^{'} & - i \bar{g^{'}} / 2 & * \end{matrix}| = \frac{1}{2} |\begin{matrix} f^{'} & 0 & * \\ 0 & \bar{f^{'}} & * \\ g^{'} & \bar{g^{'}} & * \end{matrix}|,

where we added the second column times −i to the first one, then added the first column times −i/2 to the second one, and then factored 2 from the first column, −i from the second one, and 1/2 from the third row.

Now, let f_j(z₁, z₂, z₃, z₄), j ∈ {1, 2, 3, 4, 5}, be analytic functions in each variable z_i = x_i + iy_i. We would like to compute the Jacobian of the following system of real-valued functions of x₁, y₁, …, x₄, y₄:

R e (f_{1}), I m (f_{1}), R e (f_{2}), I m (f_{2}), R e (f_{3}), I m (f_{3}), R e (f_{4}), R e (f_{5}) .

(5.11)

That is, we are interested in

J a c = |\begin{matrix} R e (f_{11}) & - I m (f_{11}) & R e (f_{12}) & - I m (f_{12}) & R e (f_{13}) & - I m (f_{13}) & R e (f_{14}) & - I m (f_{14}) \\ I m (f_{11}) & R e (f_{11}) & I m (f_{12}) & R e (f_{12}) & I m (f_{13}) & R e (f_{13}) & I m (f_{14}) & R e (f_{14}) \\ R e (f_{21}) & - I m (f_{21}) & R e (f_{22}) & - I m (f_{22}) & R e (f_{23}) & - I m (f_{23}) & R e (f_{24}) & - I m (f_{24}) \\ I m (f_{21}) & R e (f_{21}) & I m (f_{22}) & R e (f_{22}) & I m (f_{23}) & R e (f_{23}) & I m (f_{24}) & R e (f_{24}) \\ R e (f_{31}) & - I m (f_{31}) & R e (f_{32}) & - I m (f_{32}) & R e (f_{33}) & - I m (f_{33}) & R e (f_{34}) & - I m (f_{34}) \\ I m (f_{31}) & R e (f_{31}) & I m (f_{32}) & R e (f_{32}) & I m (f_{33}) & R e (f_{33}) & I m (f_{34}) & R e (f_{34}) \\ R e (f_{41}) & - I m (f_{41}) & R e (f_{42}) & - I m (f_{42}) & R e (f_{43}) & - I m (f_{43}) & R e (f_{44}) & - I m (f_{44}) \\ R e (f_{51}) & - I m (f_{51}) & R e (f_{52}) & - I m (f_{52}) & R e (f_{53}) & - I m (f_{53}) & R e (f_{54}) & - I m (f_{54}) \end{matrix}|,

where $f_{j i} ≔ \partial_{z_{i}} f_{j}$ ⁠. By performing the same row and column operations as for the determinant D above, we get that

J a c = \frac{1}{2 i} |\begin{matrix} f_{11} & 0 & f_{12} & 0 & f_{13} & 0 & f_{14} & 0 \\ 0 & {\bar{f}}_{11} & 0 & {\bar{f}}_{12} & 0 & {\bar{f}}_{13} & 0 & {\bar{f}}_{14} \\ f_{21} & 0 & f_{22} & 0 & f_{23} & 0 & f_{24} & 0 \\ 0 & {\bar{f}}_{21} & 0 & {\bar{f}}_{22} & 0 & {\bar{f}}_{23} & 0 & {\bar{f}}_{24} \\ f_{31} & 0 & f_{32} & 0 & f_{33} & 0 & f_{34} & 0 \\ 0 & {\bar{f}}_{31} & 0 & {\bar{f}}_{32} & 0 & {\bar{f}}_{33} & 0 & {\bar{f}}_{34} \\ f_{41} & {\bar{f}}_{41} & f_{42} & {\bar{f}}_{42} & f_{43} & {\bar{f}}_{43} & f_{44} & {\bar{f}}_{44} \\ f_{51} & {\bar{f}}_{51} & f_{52} & {\bar{f}}_{52} & f_{53} & {\bar{f}}_{53} & f_{54} & {\bar{f}}_{54} \end{matrix}|

with the constant in front of the determinant coming from (i/2)³ × 2⁴ × (−i)⁴ × (1/2)², where the first factor is due to multiplications of the second, fourth, and sixth rows by −2i, the second is due to factoring 2 from the odd columns, the third one comes from factoring −i from the even columns, and the fourth factor is due to factoring 1/2 from the last two rows. Assume further that

\{\begin{cases} f_{1} (z_{1}, z_{2}, z_{3}, z_{4}) = z_{1} + z_{2} + z_{3} + z_{4}, \\ f_{2} (z_{1}, z_{2}, z_{3}, z_{4}) = z_{1} z_{2} + z_{1} z_{3} + z_{1} z_{4} + z_{2} z_{3} + z_{2} z_{4} + z_{3} z_{4}, \\ f_{3} (z_{1}, z_{2}, z_{3}, z_{4}) = z_{2} z_{3} z_{4} + z_{1} z_{3} z_{4} + z_{1} z_{2} z_{4} + z_{1} z_{2} z_{3} . \end{cases})

(5.12)

Then, by using the above explicit expressions and subtracting the first (respectively, second) column from the third, fifth, and seventh (respectively, fourth, sixth, and eighth), we get that

\begin{array}{l} J a c & = \frac{1}{2 i} |\begin{matrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ * & * & z_{1} - z_{2} & 0 & z_{1} - z_{3} & 0 & z_{1} - z_{4} & 0 \\ * & * & 0 & {\bar{z}}_{1} - {\bar{z}}_{2} & 0 & {\bar{z}}_{1} - {\bar{z}}_{3} & 0 & {\bar{z}}_{1} - {\bar{z}}_{4} \\ * & * & \frac{z_{1} - z_{2}}{{(z_{3} + z_{4})}^{- 1}} & 0 & \frac{z_{1} - z_{3}}{{(z_{2} + z_{4})}^{- 1}} & 0 & \frac{z_{1} - z_{4}}{{(z_{2} + z_{3})}^{- 1}} & 0 \\ * & * & 0 & \frac{{\bar{z}}_{1} - {\bar{z}}_{2}}{{({\bar{z}}_{3} + {\bar{z}}_{4})}^{- 1}} & 0 & \frac{{\bar{z}}_{1} - {\bar{z}}_{3}}{{({\bar{z}}_{2} + {\bar{z}}_{4})}^{- 1}} & 0 & \frac{{\bar{z}}_{1} - {\bar{z}}_{4}}{{({\bar{z}}_{2} + {\bar{z}}_{3})}^{- 1}} \\ * & * & (z_{1} - z_{2}) g_{42} & ({\bar{z}}_{1} - {\bar{z}}_{2}) {\bar{g}}_{42} & (z_{1} - z_{3}) g_{43} & ({\bar{z}}_{1} - {\bar{z}}_{3}) {\bar{g}}_{43} & (z_{1} - z_{4}) g_{44} & ({\bar{z}}_{1} - {\bar{z}}_{4}) {\bar{g}}_{44} \\ * & * & (z_{1} - z_{2}) g_{52} & ({\bar{z}}_{1} - {\bar{z}}_{2}) {\bar{g}}_{52} & (z_{1} - z_{3}) g_{53} & ({\bar{z}}_{1} - {\bar{z}}_{3}) {\bar{g}}_{53} & (z_{1} - z_{4}) g_{54} & ({\bar{z}}_{1} - {\bar{z}}_{4}) {\bar{g}}_{54} \end{matrix}|, \end{array}

where $g_{j i} (z_{1}, z_{2}, z_{3}, z_{4}) ≔ {(z_{1} - z_{i})}^{- 1} (f_{j i} - f_{j 1}) (z_{1}, z_{2}, z_{3}, z_{4})$ ⁠, j ∈ {4, 5} and i ∈ {2, 3, 4}. Hence,

J a c = \frac{| z_{1} - z_{2} |^{2} | z_{1} - z_{3} |^{2} | z_{1} - z_{4} |^{2}}{2 i} |\begin{matrix} 1 & 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 & 1 \\ z_{3} + z_{4} & 0 & z_{2} + z_{4} & 0 & z_{2} + z_{3} & 0 \\ 0 & {\bar{z}}_{3} + {\bar{z}}_{4} & 0 & {\bar{z}}_{2} + {\bar{z}}_{4} & 0 & {\bar{z}}_{2} + {\bar{z}}_{3} \\ g_{42} & {\bar{g}}_{42} & g_{43} & {\bar{g}}_{43} & g_{44} & {\bar{g}}_{44} \\ g_{52} & {\bar{g}}_{52} & g_{53} & {\bar{g}}_{53} & g_{54} & {\bar{g}}_{54} \end{matrix}| .

Absolutely analogous computation now implies that

J a c = \frac{| z_{1} - z_{2} |^{2} | z_{1} - z_{3} |^{2} | z_{1} - z_{4} |^{2} | z_{2} - z_{3} |^{2} | z_{2} - z_{4} |^{2}}{2 i} |\begin{matrix} 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ h_{43} & {\bar{h}}_{43} & h_{44} & {\bar{h}}_{44} \\ h_{53} & {\bar{h}}_{53} & h_{54} & {\bar{h}}_{54} \end{matrix}|,

where $h_{j i} (z_{1}, z_{2}, z_{3}, z_{4}) ≔ {(z_{2} - z_{i})}^{- 1} (g_{j i} - g_{j 2}) (z_{1}, z_{2}, z_{3}, z_{4})$ ⁠, j ∈ {4, 5} and i ∈ {3, 4}. The above expression immediately yields that

J a c = \frac{\prod_{i < j} | z_{i} - z_{j} |^{2}}{2 i} |\begin{matrix} k_{4} & {\bar{k}}_{4} \\ k_{5} & {\bar{k}}_{5} \end{matrix}| = \prod_{i < j} | z_{i} - z_{j} |^{2} I m (k_{4} {\bar{k}}_{5}),

(5.13)

where $k_{j} (z_{1}, z_{2}, z_{3}, z_{4}) ≔ {(z_{3} - z_{4})}^{- 1} (g_{j 4} - g_{j 3}) (z_{1}, z_{2}, z_{3}, z_{4})$ ⁠, j ∈ {4, 5}. Finally, let

w (z) ≔ \sqrt{(z - z_{1}) (z - z_{2}) (z - z_{3}) (z - z_{4})}

be a branch such that $w (z) = z^{2} + O (z)$ as z → ∞ with branch cuts γ₁₂ and γ₃₄ that are bounded, disjoint, and smooth and where γ_ij connects z_i to z_j. Furthermore, select a smooth arc γ₃₂ disjoint (except for the endpoints) from the previous two. Set

f_{4} (z_{1}, z_{2}, z_{3}, z_{4}) ≔ 4 \int_{γ_{12}} w (z) d z and f_{5} (z_{1}, z_{2}, z_{3}, z_{4}) ≔ 4 \int_{γ_{32}} w (z) d z,

(5.14)

where we integrate w(z) on the positive side of γ₁₂. Let O ⊂ {z_i ≠ z_j, i ≠ j, i, j ∈ {1, 2, 3, 4}} be a domain such that there exist arcs γ_ij(z₁, z₂, z₃, z₄) with the above properties for each (z₁, z₂, z₃, z₄) ∈ O, which, in addition, possess parameterizations that depend continuously on each variable z₁, z₂, z₃, z₄. Then, the functions f_j(z₁, z₂, z₃, z₄), j ∈ {4, 5}, are analytic in each variable z_i for (z₁, z₂, z₃, z₄) ∈ O. Furthermore, it can be readily computed that

g_{j i} = 2 \int \frac{w (z) d z}{(z - z_{1}) (z - z_{i})}, h_{j i} = - 2 \int \frac{w (z) d z}{(z - z_{1}) (z - z_{2}) (z - z_{i})}, and k_{j} = 2 \int \frac{d z}{w (z)},

where the integrals are taken over γ₁₂ when j = 4 and γ₃₂ when j = 5. Trivially, (5.13) can be rewritten as

J a c = 4 \prod_{i < j} | z_{i} - z_{j} |^{2} I m (\int_{γ_{12}} \frac{d z}{w (z)} \bar{\int_{γ_{32}} \frac{d z}{w (z)}}) .

(5.15)

Now, consider the Riemann surface $S ≔ \{z ≔ (z, w) : w^{2} = (z - z_{1}) (z - z_{2}) (z - z_{3}) (z - z_{4})\}$ ⁠. Denote by $π : S \to \bar{C}$ the natural projection π(z) = z and write w(z) for a rational function on $S$ such that z = (z, w(z)). Let β ≔ π⁻¹(γ₁₂) and α ≔ π⁻¹(γ₃₂). Orient these cycles so that

2 \int_{γ_{12}} \frac{d z}{w (z)} = \oint_{β} \frac{d z}{w (z)} and 2 \int_{γ_{32}} \frac{d z}{w (z)} = \oint_{α} \frac{d z}{w (z)} .

Observe that the cycles α, β form the right pair at the point of their intersection and that $S \ {α \cup β}$ is simply connected. Hence, the cycles α, β form a homology basis on $S$ ⁠. Since the genus of $S$ is 1, it has a unique (up to multiplication by a constant) holomorphic differential. It is quite easy to check that this differential is dz/w(z). Hence, we get from (5.15) that

J a c = \prod_{i < j} | z_{i} - z_{j} |^{2} I m (\oint_{β} \frac{d z}{w (z)} \bar{\oint_{α} \frac{d z}{w (z)}}) > 0

(5.16)

when (z₁, z₂, z₃, z₄) ∈ O, where the last inequality was shown by Riemann.

Now, let $Q (z; t) = \frac{1}{4} (z - a_{1} (t)) (z - b_{1} (t)) (z - a_{2} (t)) (z - b_{2} (t))$ be the polynomial from Theorem 2.1. It can be easily deduced from (2.6) that

\{\begin{matrix} f_{1} (a_{1}, b_{1}, a_{2}, b_{2}) & = & 0, \\ f_{2} (a_{1}, b_{1}, a_{2}, b_{2}) & = & - 2 t, \\ f_{3} (a_{1}, b_{1}, a_{2}, b_{2}) & = & - 4, \end{matrix})

(5.17)

where the functions f_i(z₁, z₂, z₃, z₄), i ∈ {1, 2, 3}, are given by (5.12).

Fix $t^{*} \in O_{two−cut}$ and let δ^* > 0 be small enough so that all four disks ${| z - z_{i}^{*} | \leq δ^{*}}$ are disjoint, where $z_{1}^{*} = a_{1} (t^{*})$ ⁠, $z_{2}^{*} = b_{1} (t^{*})$ ⁠, $z_{3}^{*} = a_{2} (t^{*})$ ⁠, and $z_{4}^{*} = b_{2} (t^{*})$ ⁠. Let $O ≔ {(z_{1}, z_{2}, z_{3}, z_{4}) : | z_{i} - z_{i}^{*} | < δ^{*}}$ ⁠, $s_{i}^{*}$ be the point of intersection of ${| z - z_{i}^{*} | = δ^{*}}$ and $J_{t^{*}}$ ⁠, i ∈ {1, 2, 3, 4}, and $u_{i}^{*}$ be the point of intersection of ${| z - z_{i}^{*} | = δ^{*}}$ and Γ(b₁(t^*), a₂(t^*)), i ∈ {2, 3}, where, as usual, Γ(a, b) is the subarc of the trajectories of ϖ_t(z) connecting a and b. Then, we can choose $γ_{12} = [z_{1}, s_{1}^{*}] \cup Γ (s_{1}^{*}, s_{2}^{*}) \cup [s_{2}^{*}, z_{2}]$ ⁠, $γ_{32} = [z_{3}, u_{3}^{*}] \cup Γ (u_{3}^{*}, u_{2}^{*}) \cup [u_{2}^{*}, z_{2}]$ ⁠, and $γ_{34} = [z_{3}, s_{3}^{*}] \cup Γ (s_{3}^{*}, s_{4}^{*}) \cup [s_{4}^{*}, z_{4}]$ ⁠, where [a, b] is the line segment connecting a and b in $C$ (see Fig. 11). Clearly, the arcs γ_ij continuously depend on (z₁, z₂, z₃, z₄) ∈ O. Now, the relations (5.9) can be rewritten as

R e (f_{4} (a_{1}, b_{1}, a_{2}, b_{2})) = 0 and R e (f_{5} (a_{1}, b_{1}, a_{2}, b_{2})) = 0

(5.18)

with f_j(z₁, z₂, z₃, z₄), j ∈ {4, 5}, given by (5.14), where we set w(z) ≔ Q^1/2(z; t). It follows from (5.16) and the implicit function theorem that there exists a neighborhood of t^* in which system (5.17) and (5.18) is uniquely solvable and the solution, say, $(a_{1}^{*} (t), b_{1}^{*} (t), a_{2}^{*} (t), b_{2}^{*} (t))$ ⁠, is such that the real and imaginary parts of $a_{i}^{*} (t), b_{i}^{*} (t)$ are real analytic functions of Re(t) and Im(t) for t in this neighborhood. These local solutions are unique only locally around the point (a₁(t^*), b₁(t^*), a₂(t^*), b₂(t^*)), and we still need to argue that they do coincide with the zeros a_i(t), b_i(t) of Q(z; t) [of course, it holds that $a_{i}^{*} (t^{*}) = a_{i} (t^{*})$ and $b_{i}^{*} (t^{*}) = b_{i} (t^{*})$ ].

FIG. 11.

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A schematic illustration of γ_ij and $z_{i}^{*}$ ⁠.

In what follows, we always assume that t belongs to a disk centered at t^* of small enough radius so that the functions $a_{i}^{*} (t), b_{i}^{*} (t)$ are defined and continuous in this disk. Let

Q^{*} (z; t) ≔ \frac{1}{4} (z - a_{1}^{*} (t)) (z - b_{1}^{*} (t)) (z - a_{2}^{*} (t)) (z - b_{2}^{*} (t)) and ϖ_{t}^{*} (z) ≔ - Q^{*} (z; t) d z^{2} .

(5.19)

Furthermore, let $U^{*} (z; t)$ be defined as in (2.8) with Q(z; t) replaced by Q^*(z; t). For the moment, choose the branch cut for Q^*(z;t)^1/2 as in the paragraph between (5.17) and (5.18). Each function $U^{*} (z; t)$ is harmonic off the chosen branch cut and can be continued harmonically across it by ${- U}^{*} (z; t)$ ⁠. Moreover, it follows immediately from their definition that the functions $U^{*} (z; t)$ are uniformly bounded above and below on any compact set for all considered values of the parameter t. Thus, they converge to $U (z; t^{*})$ locally uniformly in $C \ {a_{1} (t^{*}), b_{1} (t^{*}), a_{2} (t^{*}), b_{2} (t^{*})}$ as t → t*. Since the critical graph of $ϖ_{t}^{*} (z)$ is the zero-level set of $U^{*} (z; t)$ ⁠, it converges to the critical graph of $ϖ_{t^{*}} (z)$ in any disk {|z| < R}. Due to relations (5.18), the argument at the beginning of Sec. V C also shows that the critical graph of $ϖ_{t}^{*} (z)$ has three short critical trajectories, which, due to uniform convergence, necessarily connect $a_{1}^{*} (t)$ to $b_{1}^{*} (t)$ ⁠, $b_{1}^{*} (t)$ to $a_{2}^{*} (t)$ ⁠, and $a_{2}^{*} (t)$ to $b_{2}^{*} (t)$ (the disk around t^* can be decreased if necessary). Thus, arguing as in Sec. V C and using uniform convergence, we can show that Fig. 6 also schematically represents the critical and critical orthogonal graphs of $ϖ_{t}^{*} (z)$ ⁠. Moreover, let us now take the branch cut for Q^*(z;t)^1/2, say, $J_{t}^{*}$ ⁠, along the short critical trajectories of $ϖ_{t}^{*} (z)$ connecting $a_{i}^{*} (t)$ to $b_{i}^{*} (t)$ ⁠. Then, the shading on Fig. 6 corresponds to regions where $U^{*} (z; t)$ is positive (white) and negative (gray).

Define $Γ_{t}^{*}$ to be the union of the critical orthogonal trajectory of $ϖ_{t}^{*} (z)$ that connects infinity to $a_{1}^{*} (t)$ ⁠, its short critical trajectories, and the critical orthogonal trajectory that connects $b_{2}^{*} (t)$ to infinity. Orient it so that the positive direction proceeds from $a_{1}^{*} (t)$ to $b_{2}^{*} (t)$ ⁠. Let the measures $μ_{t}^{*}$ be given by (2.7) with Q(z; t) replaced by Q^*(z; t) and J_t replaced by $J_{t}^{*}$ ⁠. Clearly, each $μ_{t}^{*}$ is a positive measure. Moreover, it has a unit mass by the Cauchy theorem and since $Q^{*} {(z; t)}^{1 / 2} = (z^{2} - t) / 2 + 1 / z + O (1 / z^{2})$ due to (5.17) [see also (5.3)]. Thus, it holds that

F^{*} (z; t) ≔ Q^{*} {(z; t)}^{1 / 2} + \frac{V^{'} (z; t)}{2} - \int \frac{d μ_{t}^{*} (s)}{z - s} = O (z^{- 2})

as z → ∞ and F^*(z; t) is holomorphic in $\bar{C} \ J_{t}^{*}$ ⁠. It follows from the well-known behavior of Cauchy integrals of smooth densities [see Ref. 36 (Sec. 1.8)] that the traces of F^*(z; t) on $J_{t}^{*}$ are bounded. It further follows from the Sokhotski–Plemelj formulas [see Ref. 36 (Sec. 1.4)] that

\begin{array}{l} F_{+}^{*} (s; t) - F_{-}^{*} (s; t) \\ = Q_{+}^{*} {(s; t)}^{1 / 2} - Q_{-}^{*} {(s; t)}^{1 / 2} - {(\int \frac{Q_{+}^{*} {(w; t)}^{1 / 2}}{w - z} \frac{d w}{π i})}_{+} + {(\int \frac{Q_{+}^{*} {(w; t)}^{1 / 2}}{w - z} \frac{d w}{π i})}_{-} \\ = Q_{+}^{*} {(s; t)}^{1 / 2} - Q_{-}^{*} {(s; t)}^{1 / 2} - 2 Q_{+}^{*} {(s; t)}^{1 / 2} \equiv 0 \end{array}

for $s \in J_{t}^{*}$ ⁠. Hence, F^*(z; t) is an entire function and, therefore, is identically zero. This observation, in particular, yields that

U^{*} (z; t) ≔ R e (2 \int_{b_{2}^{*} (t)}^{z} Q^{*} {(s; t)}^{1 / 2} d s) = ℓ_{t}^{*} - R e (V (z; t)) - 2 U^{μ_{t}^{*}} (z)

for some constant $ℓ_{t}^{*}$ [see also (2.8)]. Since $U^{*} (z; t)$ can be harmonically continued across $J_{t}^{*}$ by ${- U}^{*} (z; t)$ ⁠, we get that $μ_{t}^{*}$ satisfies (2.4); that is, $J_{t}^{*}$ has the S-property in the field Re(V(z; t)). Since $Γ_{t}^{*} \in T$ ⁠, it follows from the uniqueness part of Theorem 2.1(2) that $μ_{t}^{*} = μ_{t}$ ⁠. In particular, $a_{i}^{*} (t) = a_{i} (t)$ and $b_{i}^{*} (t) = b_{i} (t)$ ⁠, i ∈ {1, 2}.

Since any compact subset of $O_{two−cut}$ can be covered by finitely many disks where the above considerations hold, the functions a_i(t), b_i(t) continuously depend on $t \in O_{two−cut}$ ⁠, and moreover, their real and imaginary parts are real analytic functions of Re(t) and Im(t).

Let us now show that at no point in $O_{two−cut}$ are the functions a_i(t), b_i(t) analytic in t. Suppose, for the sake of contradiction, that one of these functions, say, b₂(t), is analytic in t at some point $t_{0} \in O_{two−cut}$ ⁠. Then, all the other endpoint functions, a₁(t), b₁(t), and a₂(t), are analytic in t at t₀ as well. Indeed, from (5.12) and (5.17), we have that

\{\begin{aligned} a_{1} + b_{1} + a_{2} = - b_{2}, \\ a_{1} b_{1} + a_{1} a_{2} + b_{1} a_{2} = - 2 t - b_{2} (a_{1} + b_{1} + a_{2}) = - 2 t + b_{2}^{2}, \\ a_{1} b_{1} a_{2} = - 4 - b_{2} (a_{1} b_{1} + a_{1} a_{2} + b_{1} a_{2}) = - 4 - b_{2} (- 2 t + b_{2}^{2}) = - 4 + 2 t b_{2} - b_{2}^{3} . \end{aligned})

Hence, a₁(t), b₁(t), a₂(t) are roots of the cubic equation

z^{3} + b_{2} z^{2} + (- 2 t + b_{2}^{2}) z + (4 - 2 t b_{2} + b_{2}^{3}) = 0 .

Since a₁(t), b₁(t), a₂(t) are pairwise different, they analytically depend on the coefficients of the cubic equation. That is, they are analytic functions of t and b₂. This implies that a₁(t), b₁(t), and a₂(t) are analytic in t at t₀. The analyticity of a₁(t), b₁(t), a₂(t), and b₂(t) yields that the integral

\frac{1}{π i} \int_{Γ_{t} [a_{1} (t), b_{1} (t)]} Q_{+}^{1 / 2} (s; t) d s

is also an analytic function of t at t₀. However, the integral above is equal to −ω(t) [see (4.6)], which is a real number [see (2.7)]. Hence, ω(t) must be constant in U. On the other hand, it follows from (3.6), the proof of which in Subsection V E is independent of the current considerations, that ω(t) is not constant in $O_{two−cut}$ ⁠. We also have already shown that ω(t) is real analytic in Re(t) and Im(t) and, therefore, cannot be locally constant. This contradiction proves that the endpoints a₁(t), b₁(t), a₂(t), and b₂(t) are not analytic functions of t at any point in $O_{two−cut}$ ⁠.

E. Proof of (3.6)

It follows from Ref. 37 (Theorem 5.11) that if t remains in a bounded set, so do the zeros of Q(z; t). Fix $t^{*} \in \partial O_{two−cut}$ ⁠, and let {t_m} be a sequence such that t_m → t^{^*} as m → ∞. Restricting to a subsequence if necessary, we see that there exist $a_{i}^{*}, b_{i}^{*}$ such that e(t_m) → e^* as m → ∞, where e ∈ {a₁, b₁, a₂, b₂}. Clearly, polynomials Q(z; t_m) converge uniformly on compact subsets of $C$ to Q^*(z), the polynomial with zeros $a_{i}^{*}, b_{i}^{*}$ and leading coefficient 1/4. Let ϖ^*(z) ≔ −Q^*(z)dz². Then, repeating the argument after (5.19), we get that the critical graphs of $ϖ_{t_{m}} (z)$ converge to the critical graph of ϖ^*(z) in any disk |z| < R. Furthermore, as stated in Ref. 28 (Theorem 3.3), there exists R_t > 0 such that every trajectory of ϖ_t(z) entering {|z| > R_t} necessarily remains in {|z| > R_t} and tends to infinity. Moreover, examination of the proof of Ref. 28 (Theorem 3.3) also shows that one can take $R_{t} = 2^{- 1 / 2} \max_{i} \{| a_{i} (t) |, | b_{i} (t) |\}$ and that if a trajectory enters S_{(2k−1)π/3,δ} ∩ {|z| > R_t}, k ∈ {0, 1, 2}, it stays in this sector [see (5.10)]. Put $R^{*} ≔ \sup_{m} R_{t_{m}}$ ⁠. As all the differentials $ϖ_{t_{m}} (z)$ have structurally “the same” critical graph (see Fig. 6) and all the trajectories of $ϖ_{t_{m}} (z)$ and ϖ^*(z) entering S_{(2k−1)π/3,δ} ∩{|z| > R^*} must remain there, tend to infinity, and not intersect, the unbounded critical trajectories of ϖ^*(z) behave like on Fig. 6.

Observe that the first and the third equations of (5.17) [see also (5.12)] must remain true for $a_{1}^{*}, b_{1}^{*}, a_{2}^{*}, b_{2}^{*}$ as well. Hence, we cannot simultaneously have that $a_{1}^{*} = b_{1}^{*}$ and $a_{2}^{*} = b_{2}^{*}$ ⁠. As short trajectories cannot form loops and their tangent vectors cannot become parallel at each critical point, the above considerations yield that the critical and, therefore, critical orthogonal graphs of ϖ^*(t) look like either on Fig. 6 or on Fig. 5 or as the graphs obtained by reflection across the line L_2π/3 of the graphs on Fig. 5. Repeating the arguments at the end of Sec. V D, we see that ϖ^*(t) gives rise to an S-contour in $T$ ⁠. The limits in (3.6) now follow from the uniqueness of such a S-contour and Theorem 3.1 [in particular, the critical graph ϖ_*(t) cannot look like as on Fig. 6].

VI. FUNCTIONS $D (z; t)$ AND $Q (z; t)$

In this section, we prove Propositions 4.1 and 4.2 as well as discuss other properties of $Q (z; t)$ ⁠. We consider the parameter $t \in O_{two−cut}$ to be fixed and stop indicating the dependence on t of the various quantities appearing below whenever this does not introduce ambiguity and is convenient.

A. Proof of Proposition 4.1

It follows from (1.11) and (2.6) and the choice of the branch of Q^1/2(z) that

Q^{1 / 2} (z) = \frac{z^{2} - t}{2} + \frac{1}{z} + \frac{μ_{1}}{z^{2}} + O (\frac{1}{z^{3}}) .

(6.1)

It further follows from the choice of the constant ς, d₁ in (4.1) that

z + \int_{I_{t}} \frac{3 ς}{s - z} \frac{d s}{Q^{1 / 2} (s)} = z - \frac{2 t}{z} - \frac{3 ς d_{1}}{z^{2}} - \frac{3 ς d_{2}}{z^{3}} + O (\frac{1}{z^{4}})

as z → ∞. Thus, the analyticity properties of $D (z)$ as well as (4.4) now follow from the fact that the product of the above functions behaves like

- \frac{3 V (z)}{2} + 1 - \frac{3 ς d_{1}}{2} + \frac{t^{2} + μ_{1} - 3 ς d_{2} / 2}{z} + O (\frac{1}{z^{2}})

as z → ∞. Moreover, since $Q_{+}^{1 / 2} (s) = - Q_{-}^{1 / 2} (s)$ for s ∈ J_t, we get the first relation in (4.3). The second relation in (4.3) follows from the Plemelj–Sokhotski formula

{(\int_{I_{t}} \frac{ς}{s - z} \frac{d s}{Q^{1 / 2} (s)})}_{+} - {(\int_{I_{t}} \frac{ς}{s - z} \frac{d s}{Q^{1 / 2} (s)})}_{-} = \frac{2 π i ς}{Q^{1 / 2} (z)}, z \in I_{t} ° .

B. Proof of Proposition 4.2

Since the arc I_t is homologous to the short critical trajectory of −Q(z)dz² connecting b₁ and a₂ and J_t,1 = Γ[a₁, b₁] is such a trajectory (see Fig. 6 and Conventions 1 and 2), the constants τ, ω are indeed real. Let

g (z) ≔ \int \log (z - s) d μ_{t} (s), z \in C \ Γ (e^{π i} \infty, b_{2}],

(6.2)

where we take the principal branch of log(· − s) holomorphic outside of $Γ (e^{π i} \infty, s]$ and μ_t is the equilibrium measure defined in (2.7). It follows directly from definition (6.2) that

\partial_{z} g (z) = \int \frac{d μ_{t} (s)}{z - s},

where, as usual, ∂_z ≔ (∂_x − i∂_y)/2. Therefore, it can be deduced from (2.2) and (2.6) that

g (z) = \frac{V (z) - ℓ_{*}}{2} + \int_{b_{2}}^{z} Q^{1 / 2} (s) d s = \frac{V (z) - ℓ_{*}}{2} + Q (z),

(6.3)

where, as usual, we take the branch $Q^{1 / 2} (z) = \frac{1}{2} z^{2} + O (z)$ ⁠, ℓ_* is a constant such that the equality holds at b₂ [note that Re(ℓ_*) = ℓ; see (2.2)], and $Q (z)$ is given by (4.5). Property (4.7) clearly follows from (6.2) and (6.3). In view of (6.3), let us define

ϕ_{e} (z) ≔ 2 \int_{e}^{z} Q^{1 / 2} (s) d s, e \in {a_{1}, b_{1}, a_{2}, b_{2}},

(6.4)

holomorphically in $C \ Γ (e^{π i} \infty, b_{2}]$ when e = b₂, in $C \ Γ [a_{1}, e^{π i / 3} \infty)$ when e = a₁, and in $C \ (Γ (e^{π i} \infty, b_{1}) \cup Γ (a_{2}, e^{π i / 3} \infty))$ when e ∈ {b₁, a₂}. Clearly, $ϕ_{b_{2}} (z) = 2 Q (z)$ ⁠. One can readily check that

ϕ_{b_{2}} (z) = \{\begin{cases} ϕ_{a_{2}} (z) \pm 2 π i (1 - ω), \\ ϕ_{b_{1}} (z) \pm 2 π i (1 - ω) + 2 π i τ, \\ ϕ_{a_{1}} (z) \pm 2 π i + 2 π i τ, \end{cases}) z \in C \ Γ,

(6.5)

where the plus sign is used if z lies to the left of Γ and the minus sign if z lies to the right of it, and

ϕ_{b_{2} \pm} (s) = \{\begin{cases} \pm 2 π i μ_{t} (Γ [s, b_{2}]), & s \in Γ (a_{2}, b_{2}), \\ \pm 2 π i μ_{t} (Γ [s, b_{2}]) + 2 π i τ, & s \in Γ (a_{1}, b_{1}) . \end{cases})

(6.6)

The jump relations in (4.8) now easily follow from (6.5) and (6.6).

For future use, let us record that (6.3), (6.5), and (6.6) imply that

g_{+} (s) - g_{-} (s) = \{\begin{matrix} 0, & s \in Γ (b_{2}, e^{π i / 3} \infty), \\ \pm ϕ_{b_{2} \pm} (s), & s \in Γ (a_{2}, b_{2}), \\ 2 π i (1 - ω), & s \in Γ (b_{1}, a_{2}), \\ \pm (ϕ_{b_{2} \pm} (s) - 2 π i τ), & s \in Γ (a_{1}, b_{1}), \\ 2 π i, & s \in Γ (e^{π i} \infty, a_{1}), \end{matrix})

(6.7)

and that

g_{+} (s) + g_{-} (s) - V (s) + ℓ_{*} = \{\begin{matrix} ϕ_{b_{2}} (s), & s \in Γ (b_{2}, e^{π i / 3} \infty), \\ 0, & s \in Γ (a_{2}, b_{2}), \\ ϕ_{a_{2}} (s), & s \in Γ (b_{1}, a_{2}), \\ 2 π i τ, & s \in Γ (a_{1}, b_{1}), \\ ϕ_{a_{1}} (s) + 2 π i τ, & s \in Γ (e^{π i} \infty, a_{1}) . \end{matrix})

(6.8)

C. Local analysis at e ∈ {a₁, b₁, a₂, b₂}

Given e ∈ {a₁, b₁, a₂, b₂}, let

U_{e} ≔ \{z : | z - e | < δ_{e} ρ (t) / 3\},

(6.9)

where δ_e ∈ (0, 1] to be adjusted later, and we shall specify the function ρ(t) at the end of this subsection. Set

J_{e} ≔ U_{e} \cap J_{t} and I_{e} ≔ U_{e} \cap (Γ_{t} \ J_{t}),

(6.10)

where the arcs J_e and I_e inherit their orientation from Γ_t, and we assume that the value of ρ(t) is small enough so that these arcs are connected and that I_e is a subarc of the orthogonal critical trajectory of −Q(z)dz² emanating from e (see the remarks about Γ_t at the very beginning of Sec. IV). The latter fact and Theorem 3.2 yield that

ϕ_{e} (s) < 0, s \in I_{e}

(6.11)

(see Fig. 6). In fact, the same reasoning shows that (6.11) holds not only on I_e, but on Γ(e^πi∞, a₁) when e = a₁, on Γ(b₂, e^πi/3∞) when e = b₂, and for Re(ϕ_e(z)) on Γ(b₁, a₂) when e ∈ {b₁, a₂} (observe that these functions are also monotone on the respective arcs). Furthermore, each function ϕ_e(z) is analytic in U_e\J_e, and its traces on J_e satisfy

ϕ_{e \pm} (s) = \pm 2 π i ν_{e} μ (J_{s, e}) = 2 π e^{\pm 3 π i ν_{e} / 2} μ (J_{s, e}),

(6.12)

where J_s,e is the subarc of J_e with endpoints e and s,

ν_{e} ≔ \{\begin{matrix} 1, & e \in {b_{1}, b_{2}}, \\ - 1, & e \in {a_{1}, a_{2}}, \end{matrix})

(6.13)

and the second equality follows from (2.7) and (6.4). Since |ϕ_e(z)| ∼ |z − e|^3/2 as z → e, it follows from (6.11) and (6.12) that we can define an analytic branch of ${(- ϕ_{e})}^{2 / 3} (z)$ in U_e that is positive on I_e and satisfies ${(- ϕ_{e})}^{2 / 3} (s) = - {(2 π μ_{t} (J_{s, e}))}^{2 / 3}$ ⁠, s ∈ J_e. Since ${(- ϕ_{e})}^{2 / 3} (z)$ has a simple zero at e, it is conformal in U_e for all radii small enough. Altogether, ${(- ϕ_{e})}^{2 / 3} (z)$ maps e into the origin, is conformal in U_e, and satisfies

\{\begin{matrix} {(- ϕ_{e})}^{2 / 3} (J_{e}) & \subset & (- \infty, 0), \\ {(- ϕ_{e})}^{2 / 3} (I_{e}) & \subset & (0, \infty) . \end{matrix})

(6.14)

Furthermore, if we define ${(- ϕ_{e})}^{1 / 6} (z)$ to be holomorphic in U_e\J_e and positive on I_e, then

{(- ϕ_{e})}_{+}^{1 / 6} (s) = ν_{e} i {(- ϕ_{e})}_{-}^{1 / 6} (s), s \in J_{e} .

(6.15)

To specify ρ(t), let ρ_e(t) be the radius of the largest disk around e for which J_e, I_e are connected and in which ${(- ϕ_{e})}^{2 / 3} (z)$ is conformal. Observe that the disk around e of radius ρ_e(t) cannot contain other endpoints of J_t besides e. We set ρ(t) ≔ min_e{ρ_e(t)}. Then, the disks U_e in (6.9) are necessarily disjoint. Observe also that ρ(t) is non-zero for all $t \in O_{two−cut}$ and continuously depends on t due to continuous dependence on t of ϕ_e(z), which in itself follows from Theorem 3.2 and (6.4).

VII. FUNCTIONS Θ_n(z; t)

In this section, we prove Propositions 4.3–4.6 as well as discuss some related results. As in Sec. VI, we omit indicating the explicit dependence on t whenever convenient.

A. Riemann surface

In this subsection, we discuss properties of the Riemann surface $S$ that has already appeared between (5.15) and (5.16). Once again, set

S ≔ \{z ≔ (z, w) : w^{2} = Q (z)\} .

(7.1)

We denote by $π : S \to \bar{C}$ the natural projection π(z) = z and by ·^* a holomorphic involution on $S$ acting according to the rule z^* = (z, −w). We use notation z, s, a for points on $S$ with natural projections z, s, a.

The function w(z), defined by w²(z) = Q(z), is meromorphic on $S$ with simple zeros at the ramification points E_t = {a₁, b₁, a₂, b₂} = π⁻¹(E_t), double poles at the points on top of infinity, and is otherwise non-vanishing and finite. Put

Δ ≔ π^{- 1} (J_{t}) and S = D^{(0)} \cup Δ \cup D^{(1)},

where the domains D^(k) project onto $\bar{C} \ J_{t}$ with labels chosen so that $2 w (z) = {(- 1)}^{k} z^{2} + O (z)$ as z approaches the point on top of infinity within D^(k). For $z \in \bar{C} \ J_{t}$ ⁠, we let z^(k) stand for z ∈ D^(k) with, as agreed, π(z) = z. We define a homology basis on $S$ in the following way: we let

α ≔ π^{- 1} (I_{t}) and β ≔ π^{- 1} (J_{t, 1}),

where α is oriented toward b₁ within D⁽⁰⁾ and β is oriented so that α, β form the right pair at b₁ (see Figs. 12 and 13). It also will be convenient to put $S_{α, β} ≔ S \ {α \cup β}$ ⁠, which is simply connected.

FIG. 12.

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Schematic plot of the Riemann surface $S$ and the cycles α and β.

FIG. 13.

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Planar model of the Riemann surface $S$ and the cycles α and β.

The surface $S$ has genus one. Thus, there exists a unique holomorphic differential on $S$ normalized to have a unit period on α, say, $H$ ⁠. In fact, it can be explicitly expressed as

H (z) = {(\oint_{α} \frac{d s}{w (s)})}^{- 1} \frac{d z}{w (z)} .

(7.2)

With this notation, the choice of the homology basis, and the definition of w(z), (4.9) becomes

B = \oint_{β} H, I m (B) > 0,

(7.3)

where the last inequality is a classical result of Riemann (see Ref. 38).

Given the normalized holomorphic differential, we define

a (z) ≔ \int_{b_{2}}^{z} H,

(7.4)

where we restrict z as well as the path of integration to $S_{α, β}$ ⁠. The function $a (z)$ is a holomorphic function in $S_{α, β}$ with continuous traces on α, β (away from the point of their intersection) that satisfy

a_{+} (s) - a_{-} (s) = \{\begin{matrix} - B, & s \in α \ {b_{1}}, \\ 1, & s \in β \ {b_{1}}, \end{matrix})

(7.5)

by the normalization of $H$ and the definition of $B$ ⁠. Moreover, it continuously extends to $\partial S_{α, β}$ ⁠, the topological boundary of $S_{α, β}$ (the rectangle on Fig. 13). Note that

τ = \frac{1}{2 π i} \oint_{α} w (s) d s and ω = - \frac{1}{2 π i} \oint_{β} w (s) d s,

(7.6)

where τ, ω are the constants from (4.6). It readily follows from (7.5) and (7.6) that

\oint_{\partial S_{α, β}} (w a) (s) d s = \oint_{β} w (s) (a_{+} - a_{-}) (s) d s + \oint_{α} w (s) (a_{+} - a_{-}) (s) d s = - 2 π i (ω + B τ),

where $\partial S_{α, β}$ is oriented so that $S_{α, β}$ remains on the left when $\partial S_{α, β}$ is traversed in the positive direction. On the other hand, the function $(w a) (z)$ is meromorphic in $S_{α, β}$ with only two singularities, both polar, at ∞⁽⁰⁾ and ∞⁽¹⁾. Moreover, since w(z^*) = −w(z) and $a (z^{*}) = - a (z)$ ⁠, the residues at those poles coincide. Therefore, it holds that

ω + B τ = - \frac{1}{2 π i} \oint_{\partial S_{α, β}} (w a) (s) d s = - 2 {r e s}_{z = \infty^{(0)}} (w a) (z) .

Recall that $w (z^{(0)}) = Q^{1 / 2} (z)$ ⁠, and therefore, it has expansion (6.1) as z⁽⁰⁾ → ∞⁽⁰⁾. This and (7.4) also allow us to deduce that

a (z^{(0)}) = a (\infty^{(0)}) + \int_{\infty^{(0)}}^{z^{(0)}} H = a (\infty^{(0)}) - {(\oint_{α} \frac{d s}{w (s)})}^{- 1} (\frac{2}{z} + \frac{2 t}{3 z^{3}} + O (\frac{1}{z^{4}}))

as z⁽⁰⁾ → ∞⁽⁰⁾. Therefore, taking into account (4.1) with the definition of α and the symmetry $a (z^{*}) = - a (z)$ ⁠, we get that

ς + ω + B τ = ς + 2 a (\infty^{(0)}) + \frac{4 t}{3} {(\oint_{α} \frac{d s}{w (s)})}^{- 1} = 2 \int_{b_{2}}^{\infty^{(0)}} H = \int_{\infty^{(1)}}^{\infty^{(0)}} H,

(7.7)

where the path of integration lies entirely in $S_{α, β}$ and is involution-symmetric.

B. Jacobi inversion problem

Denote by $J a c (S) ≔ C / {Z + B Z}$ the Jacobi variety of $S$ ⁠. We shall represent elements of $J a c (S)$ as equivalence classes $[s] = {s + j + m B : j, m \in Z}$ ⁠, where $s \in C$ ⁠. One can readily deduce from (7.5) that $a (z)$ ⁠, defined in (7.4), is essentially a holomorphic bijection from $S$ onto $J a c (S)$ ⁠. Formally, we define this bijection, known as Abel’s map, by

z \in S \mapsto [\int_{b_{2}}^{z} H] \in J a c (S) .

(7.8)

Given $s \in C$ ⁠, we denote by z_[s] the unique point in $S$ such that $[\int_{b_{2}}^{z_{[s]}} H] = [s]$ ⁠.

Proposition 7.1.

Let τ, ω be given by (4.6) and ς by (4.1). Furthermore, let {N_n} be a sequence as in Theorem 4.1. Denote by z_n,k = z_n,k(t) the unique solution

z_{[s_{n, k} (t)]}

of the Jacobi inversion problem with

s_{n, k} (t) ≔ \int_{b_{2}}^{p^{(k)}} H + (n - N_{n}) ς + (ω + B τ) n, p = p (t) ≔ \frac{b_{1} b_{2} - a_{1} a_{2}}{(b_{2} - a_{2}) + (b_{1} - a_{1})},

(7.9)

k ∈ {0, 1}. Then, for any subsequence

N_{*}

⁠, the point ∞⁽⁰⁾ is a topological limit point of

{z_{n, 1}}_{n \in N_{*}}

if and only if ∞⁽¹⁾ is a topological limit point of

{z_{n, 0}}_{n \in N_{*}}

. Moreover, it holds that [s_n,0(t; N_n)] = [s_n−1,1(t; N_n)]; that is, z_n,0(N_n) = z_n−1,1(N_n).

The behavior of the points z_n,k with respect to n can be extremely chaotic. Assuming that n − N_n is constant, it is known that if the numbers ω and τ are rational, then there exist only finitely many distinct points z_n,k; when ω and τ are irrational, all the points z_n,k are distinct, lie on a Jordan curve if 1, ω, and τ are rationally dependent, and are dense on the whole surface $S$ otherwise.³⁹

Let us also point out that the point p is always finite. Indeed, it follows from (1.11) and (2.6) that b₁ + b₂ + a₁ + a₂ = 0 [see also (5.12) and (5.17)]. Thus, if b₂ + b₁ − a₁ − a₂ = 0 were true, then we would have b₁ = −b₂ and a₁ = −a₂. In this case, a₁b₁a₂ + a₁b₁b₂ + a₁a₂b₂ + b₁a₂b₂ = 0. However, this expression must be equal to −4 according to the above references.

Proof of Proposition 7.1.

Define

γ (z) ≔ {(\frac{z - b_{2}}{z - a_{2}} \frac{z - b_{1}}{z - a_{1}})}^{1 / 4}, z \in \bar{C} \ J_{t},

(7.10)

where γ(z) is holomorphic off J_t = supp(μ_t) and the branch is chosen so that γ(∞) = 1. Furthermore, set

A (z) = \frac{γ (z) + γ^{- 1} (z)}{2} and B (z) ≔ \frac{γ (z) - γ^{- 1} (z)}{- 2 i} .

(7.11)

Observe that the function A(z) was already defined in (). The functions A(z) and B(z) are holomorphic in

\bar{C} \ J_{t}

and satisfy

A (\infty) = 1, B (\infty) = 0, and A_{\pm} (s) = \pm B_{\mp} (s), s \in J_{t} ° .

(7.12)

Note that the equation (AB)(z) = 0 can be rewritten as γ⁴(z) = 1 and has two solutions, namely, ∞ and the point p from (). In fact, unless p ∈ J_t°, it a zero of B(z). Indeed, it is enough to show that γ(p) = 1 in the latter case. Let L_i ≔ γ⁴(J_t,i), i ∈ {1, 2}, which are unbounded arcs connecting the origin to the point at infinity. Let

L \subset \bar{C} \ J_{t}

be an arc connecting the point at infinity and p. Then, γ⁴(L) is a closed curve that contains 1 and does not intersect the arcs L_i and, therefore, does not wind around the origin. Thus, analytic continuation of the principal branch of the 1/4-root from 1 along γ⁴(L) leads back to the value 1 at the point 1. However, this continuation is exactly the continuation of γ(z) from the point at infinity to p along L, which does imply that γ(p) = 1 as claimed.

It follows from (7.12) that

\{\begin{matrix} (B / A) (z), & z \in D^{(0)}, \\ - (A / B) (z), & z \in D^{(1)}, \end{matrix})

(7.13)

is a rational function on $S$ with two simple zeros ∞⁽⁰⁾ and p⁽⁰⁾ and two simple poles ∞⁽¹⁾ and p⁽¹⁾ [if it happens that p ∈ J_t°, then we choose $p^{(0)} \in S$ precisely in such a way that it is a zero of (7.13) and so p⁽¹⁾ is a pole of (7.13); it is, of course, still true that these points are distinct and $π (p^{(k)}) = p$ ]. Therefore, Abel’s theorem yields that

[\int_{p^{(0)}}^{\infty^{(1)}} H] = [\int_{p^{(1)}}^{\infty^{(0)}} H],

(7.14)

while relations (7.9), in particular, imply that

[\int_{p^{(0)}}^{z_{n, 0}} H] = [\int_{p^{(1)}}^{z_{n, 1}} H] .

(7.15)

Let z_k be a topological limit of a subsequence ${z_{n_{i}, k}}$ ⁠. Holomorphy of the differential $H$ implies that

\int_{p^{(k)}}^{z_{n_{i}, k}} H = \int_{p^{(k)}}^{z_{k}} H + \int_{z_{k}}^{z_{n_{i}, k}} H \to \int_{p^{(k)}}^{z_{k}} H

as i → ∞, where the integral from z_k to $z_{n_{i}, k}$ is taken along the path that projects into a segment joining z_k and $z_{n_{i}, k}$ ⁠. The first claim of the proposition now follows from (7.14) and (7.15) and the unique solvability of the Jacobi inversion problem on $S$ ⁠. Observe that

[\int_{p^{(0)}}^{p^{(1)}} H] = [\int_{\infty^{(1)}}^{\infty^{(0)}} H] = [ς + ω + B τ]

(7.16)

by (7.7) and (7.14). Since

s_{n, 0} (N_{n}) = s_{n - 1, 1} (N_{n}) + \int_{p^{(1)}}^{p^{(0)}} H + ς + ω + B τ,

the second conclusion of the proposition easily follows.

C. Proof of Proposition 4.5

As we shall show further below, the functions Θ_n(z; t) from Proposition 4.4 vanish at z_n,1 when it belongs to D⁽⁰⁾ and do not vanish at all when z_n,1 does not belong to D⁽⁰⁾. Hence, the subsequences $N (ε) = N (t, ε)$ from Proposition 4.4 can be equivalently defined as

N (ε) ≔ \{n \in N : z_{n, 1} \notin D^{(0)} \cap π^{- 1} (\{| z | \geq 1 / ε\})\} .

Denote by r[s] the distance from 0 to [s], viewed as a lattice in $C$ ⁠. That is,

r [s] ≔ \min_{j, m \in Z} |s + j + B m|,

where s is any representative of [s]. Furthermore, for each r[s] > 0, choose ɛ_s > 0 small enough so that

|a (z) - a (\infty^{(0)})| < r [s] / 3, z \in U_{ε_{s}}, U_{ε} ≔ D^{(0)} \cap π^{- 1} (\{| z | > 1 / ε\}),

where we also assume that ɛ_s is small enough so that always $U_{ε_{s}} \subset S_{α, β}$ ⁠. Then,

[a (z_{1}) - a (z_{2})] = [s] \neq [0] \Rightarrow either z_{1} \notin U_{ε_{s}} or z_{2} \notin U_{ε_{s}}

(7.17)

as otherwise it must hold $0 < r [s] \leq | a (z_{1}) - a (z_{2}) | < (2 / 3) r [s]$ ⁠.

Assume that either x or y is not an integer. Let $z_{n, 1}^{-}, z_{n, 1}, z_{n, 1}^{+}$ be solutions of the Jacobi inversion problem (7.9) with N equal to n − 1, n, n + 1, respectively. Then, it follows from (7.9) that

[a (z_{n, 1}^{\pm}) - a (z_{n, 1})] = [\mp ς] .

As [±ς] ≠ [0] by assumption, the second assertion of the proposition follows from (7.17).

Assume now that both x and y are integers. It readily follows from (7.9) that the definition of z_n,k does not depend on the choice of N_n in this case. The first assertion of the proposition holds since

[a (z_{n + 1, 1}) - a (z_{n, 1})] = [ω + B τ] \neq [0],

where the last conclusion follows from the fact ω = μ_t(J_t,1) ∈ (0, 1) by the very definition in (4.6).

D. Proof of Proposition 4.6

Let k = N_n+1 − N_n. It follows from (7.9) that

[a (z_{n + 1, 1}) - a (z_{n, 1})] = [(1 - k) ς + ω + B τ] .

(7.18)

Set $K ≔ {k \in Z : r [(1 - k) ς + ω + B τ] = 0}$ ⁠. We have already shown that $r [ω + B τ] > 0$ ⁠. It also holds that $r [ς + ω + B τ] > 0$ since $[ς + ω + B τ] \neq [0]$ by (7.7) and the unique solvability of the Jacobi inversion problem. The following lemma easily follows from (7.17).

Lemma 7.2.

Assume that |n − N_n| ≤ N_*, and let $ε_{k} ≔ ε_{(1 - k) ς + ω + B τ}$ .

If $K = \emptyset$ , then at least one of the integers n, n + 1 belongs to $N (ε)$ for all $ε \leq \min_{| k | \leq 2 N_{*} + 1} ε_{k}$ .
If $K = {k^{'}}$ , there exists an infinite subsequence {n_l} such that $N_{n_{l} + 1} - N_{n_{l}} \neq k^{'}$ ⁠, and therefore, at least one of the integers n_l, n_l + 1 belongs to $N (ε)$ for all $ε \leq \min_{| k | \leq 2 N_{*} + 1, k \neq k^{'}} ε_{k}$ .
If there exists an infinite subsequence {n_l} such that $N_{n_{l} + 1} - N_{n_{l}} \in {0, 1}$ , then at least one of the integers n_l, n_l + 1 belongs to $N (ε)$ for all ɛ ≤ min{ɛ₀, ɛ₁}.

Inclusion $k \in K$ means that $[(1 - k) ς + ω + B τ] = [0]$ ⁠. Hence, both triples ω, x, 1 and τ, y, 1 are rationally dependent, $ς = x + B y$ ⁠. Assume that $k^{'}, k^{''} \in K$ ⁠, k′ ≠ k″. Then, it follows from (7.18) that $[ω + B τ] = [(k^{'} - 1) ς]$ and [(k″ − k′)ς] = [0]. The latter relation implies the first representation in (4.13), while the former gives the other two. It is easy to see in this case that $K = k^{'} + d Z$ ⁠. That is, if $K$ has at least two elements, then it is an arithmetic progression, ω, τ are rational numbers, ς has rational coordinates in the basis $1, B$ ⁠, and the second and third relations of (4.13) must be satisfied. Thus, we can claim the following lemma:

Lemma 7.3.

If one of the triples ω, x, 1 or τ, y, 1 is rationally independent, then $K = \emptyset$ . If not all numbers ω, τ, x, y are rational or they all rational but the second and third relations of (4.13) do not hold, then either $K = \emptyset$ or $K = {k^{'}}$ .

Assume now that all three relations of (4.13) take place. That is, $[ω + B τ] = [(k - 1) ς]$ and [dς] = [0] for some integers k, d. It follows from (7.14) that

\int_{b_{2}}^{p^{(1)}} H = \frac{1}{2} \int_{p^{(0)}}^{p^{(1)}} H = \frac{1}{2} (\int_{\infty^{(1)}}^{\infty^{(0)}} H + j + B m) = \int_{b_{2}}^{\infty^{(0)}} H + \frac{j + B m}{2}

for some $j, m \in Z$ ⁠, where we use involution-symmetric paths of integration. Note that j, m cannot be simultaneously even as this would contradict unique solvability of the Jacobi inversion problem. In fact, it holds that

[\int_{p^{(1)}}^{\infty^{(0)}} H] = [\int_{b_{2}}^{\infty^{(0)}} H + \int_{b_{2}}^{p^{(0)}} H] = [\int_{b_{2}}^{b_{1}} H] = [\frac{1 + B}{2}],

(7.19)

i.e., both j, m must be odd. Indeed, the first equality follows from the symmetry $a (z^{*}) = - a (z)$ ⁠. Next, observe that the function ${(- 1)}^{k} γ^{- 2} (z^{(k)}) - 1$ is meromorphic on $S$ with zeros at ∞⁽⁰⁾ and p⁽⁰⁾ and poles at b₁ and b₂, all simple. Hence, the second equality is a consequence of the same symmetry and Abel’s theorem. Furthermore, consider an involution-symmetric cycle represented by the anti-diagonal on Fig. 13, oriented so that it proceeds toward b₁ in D⁽⁰⁾. It is clearly homologous to α + β. Then, the symmetry of $a (z)$ and the normalization in (7.2) and (7.3) yield the final equality in (7.19). Note that the conclusion of (7.19) a posteriori holds with the bounds of integration in the left most integral of (7.19) in flipped. Therefore, adding $\int_{\infty^{(0)}}^{b_{2}} H$ to both sides of (7.9) gives us

[\int_{\infty^{(0)}}^{z_{n, 1}} H] = [\frac{1 + B}{2} + (n - N_{n}) ς + (ω + B τ) n] = [\frac{1 + B}{2} + (n k - N_{n}) ς] .

(7.20)

Since ς has rational coordinates in the basis $1, B$ with denominator d, the right-hand side of (7.20) has at most d distinct values that depend only on ρ ∈ {0, …, d − 1}, the remainder of the division of nk − N_n by d. Let z_ρ, ρ ∈ {0, …, d − 1}, be such that

[\int_{\infty^{(0)}}^{z_{ρ}} H] = [\frac{1 + B}{2} + ρ ς] .

Clearly, ${z_{n, 1}}_{n \in N} \subseteq {z_{ρ}}_{ρ = 0}^{d - 1}$ ⁠. Thus, it only remains to investigate when z_ρ = ∞⁽⁰⁾ or, equivalently, when $[(1 + B) / 2 + ρ ς] = [0]$ ⁠. From the representation of ς in (4.13), it must simultaneously hold that ρ = d(2l_j − 1)/(2i_j), j ∈ {1, 2}, for some $l_{1}, l_{2} \in Z$ ⁠. Since one of the pairs (i_j, d) is co-prime and ρ ∈ {0, …, d − 1}, this is possible only if ρ = 0 or ρ = d/2 (in the second case, of course, d must be even and i_j = 2l_j − 1 must be odd), and the former is ruled out since $[(1 + B) / 2] \neq [0]$ (or equivalently, since 0 ≠ 2l_j − 1).

Lemma 7.4.

If all three relations of (4.13) take place, then the Jacobi inversion problem (7.9) for z_n,1 has only finitely many distinct solutions and ∞⁽⁰⁾ is one of them if and only if d is even, i₁, i₂ are odd, and nk − N_n is divisible by d/2 but not by d.

E. Proof of Proposition 4.4

Recall that Abel’s map (7.8) is essentially carried out by the function $a (z)$ defined in (7.4). We shall consider the extension $\tilde{a} (z)$ of $a (z)$ to the whole surface $S$ defined by setting $\tilde{a} (s) ≔ a_{+} (s)$ for s ∈ α and s ∈ β\{b₁}. Given such an extension and (7.9), there exist unique integers j_n,k, m_n,k such that

\tilde{a} (z_{n, k}) = \tilde{a} (p^{(k)}) + (n - N_{n}) ς + (ω + B τ) n + j_{n, k} + {B m}_{n, k}, k \in {0, 1} .

(7.21)

Recall the definition of θ(u) in (4.10). The function θ(u) is holomorphic in $C$ and enjoys the following periodicity properties:

θ (u + j + B m) = \exp \{- π i {B m}^{2} - 2 π i u m\} θ (u), j, m \in Z .

(7.22)

It is also known that θ(u) vanishes only at the points of the lattice $[\frac{B + 1}{2}]$ ⁠. Let

Θ_{n, k} (z) = \exp \{- 2 π i (m_{n, k} + τ n) a (z)\} \frac{θ (a (z) - \tilde{a} (z_{n, k}) - \frac{B + 1}{2})}{θ (a (z) - \tilde{a} (p^{(k)}) - \frac{B + 1}{2})} .

(7.23)

Then, we define Θ_n(z; t) from Proposition 4.4 as well as the other functions appearing in the statement of Theorem 4.2 by

Θ_{n} (z; t) ≔ Θ_{n, 1}^{(0)} (z), ϑ_{n} (z) ≔ \frac{Θ_{n, 1}^{(0)} (z)}{Θ_{n, 1}^{(0)} (\infty)}, ϑ_{n}^{*} (z) ≔ \frac{Θ_{n, 1}^{(1)} (z)}{Θ_{n, 1}^{(0)} (\infty)}, and {\tilde{ϑ}}_{n} (z) ≔ \frac{Θ_{n, 0}^{(0)} (z)}{Θ_{n, 0}^{(1)} (\infty)},

(7.24)

where F⁽ⁱ⁾(z), i ∈ {0, 1}, stands for the pull-back under π(z) of a function F(z) from D⁽ⁱ⁾ into $\bar{C} \ J_{t}$ ⁠.

The functions Θ_n,k(z) are meromorphic on $S_{α, β}$ with exactly one pole, which is simple and located at p^(k), and exactly one zero, which is also simple and located at z_n,k (observe that these functions can be analytically continued as multiplicatively multivalued functions on the whole surface $S$ ⁠; thus, we can talk about simplicity of a pole or zero regardless whether it belongs to the cycles of a homology basis or not). Moreover, according to (7.5), (7.21), and (7.22), they possess continuous traces on α, β away from b₁ that satisfy

Θ_{n, k +} (s) = Θ_{n, k -} (s) \{\begin{matrix} \exp \{- 2 π i (ω n + (n - N_{n}) ς)\}, & s \in α \ {b_{1}}, \\ \exp \{- 2 π i τ n\}, & s \in β \ {b_{1}} . \end{matrix})

(7.25)

Recall functions A(z), B(z) from (7.11) and (7.12). To discuss boundedness properties of Θ_n,k(z) and for the asymptotic analysis in Sec. VIII, it will be convenient to define

M_{n, 0} (z) = Θ_{n, 0} (z) \{\begin{matrix} B (z), z \in D^{(0)}, \\ A (z), z \in D^{(1)}, \end{matrix}) and M_{n, 1} (z) = Θ_{n, 1} (z) \{\begin{matrix} A (z), z \in D^{(0)}, \\ - B (z), z \in D^{(1)} . \end{matrix})

(7.26)

These functions are holomorphic on $S \ {α \cup β \cup Δ}$ since the pole of Θ_n,k(z) is canceled by the zero of B(z). Each function M_n,k(z) has exactly two zeros, namely, z_n,k and ∞^(k). It follows from (7.12) and (7.25) that

\{\begin{matrix} M_{n, k \pm}^{(0)} (s) = \mp M_{n, k \mp}^{(1)} (s), & s \in J_{t, 2} °, \\ M_{n, k \pm}^{(0)} (s) = \mp e^{- 2 π i τ n} M_{n, k \mp}^{(1)} (s), & s \in J_{t, 1} °, \\ M_{n, k \pm}^{(i)} (s) = e^{{(- 1)}^{i} 2 π i (n ω + (n - N_{n}) ς)} M_{n, k \mp}^{(i)} (s), & s \in I_{t} ° . \end{matrix})

(7.27)

It further follows from (7.10) and (7.11) that |M_n,k(z)| ∼ |z − e|^−1/4 as z → e ∈ E unless z_n,k coincides with e in which case the exponent becomes 1/4. Assume now that there exists N_* ≥ 0 such that |n − N_n| ≤ N_* for all $n \in N$ ⁠. Then, for each δ > 0 there exists C(δ, N_*) independent of n such that

| M_{n, k} (z) | \leq C (δ, N_{*}), z \in O_{δ} ≔ S \ \cup_{e \in E} π^{- 1} {| z - e | < δ} .

(7.28)

Indeed, observe that {log|M_n,k(z)|} is a family of subharmonic functions in O_δ\α [the jump of M_n,k(z) is unimodular on β]. By the maximum principle for subharmonic functions, log|M_n,k(z)| reaches its maximum on ∂(O_δ\α), where the maximum is clearly finite. Since the sequence {n − N_n} is bounded by assumption and the range of $\tilde{a} (z)$ is bounded by construction, so are the sequences {m_n,k + τn} and {j_n,k + ωn} [see (7.21)], and recall that j_n,k + ωn are real and $I m (B) > 0$ ⁠. Thus, any limit point of {log|M_n,k(z)|} is obtained by taking simultaneous limit points of {n − N_n}, {m_n,k + τn}, and {j_n,k + ωn}, computing the corresponding solution z_k of the Jacobi inversion problem (7.21) and plugging all of these quantities into the right-hand side of (7.23). Hence, all these limit functions are also bounded above on the closure of O_δ\α, which proves (7.28). Finally, it holds that

|M_{n, k} (\infty^{(1 - k)})| \geq c_{ε}, n \in N (ε),

(7.29)

for some constant c_ɛ > 0 by a similar compactness argument combined with the definition of $N (ε)$ in Proposition 4.4, the observation that M_n,k(z) is non-zero at ∞^(1−k) when ∞^(1−k) ≠ z_n,k, and the last conclusion of Proposition 7.1.

F. Proof of Proposition 4.3

Define a function F(z) on $S \ {α \cup β}$ by setting $F (z^{(1)}) ≔ 1 / F (z^{(0)})$ and

F {(z^{(0)}) ≔ D}^{- 1} (z) \exp \{V (z) / 2 + Q (z)\}

[see (4.2) and (4.5)]. It follows from (4.3) and (4.8) that

F_{+} (s) = F_{-} (s) \{\begin{matrix} \exp \{2 π i (ω + ς)\}, & s \in α \ {b_{1}}, \\ \exp \{2 π i τ\}, & s \in β \ {b_{1}}, \end{matrix})

(7.30)

where one needs to recall that I_t is oriented toward a₂, while $α \cap S^{(0)}$ is oriented toward b₁. Moreover, it follows from (4.7) that

F (z^{(0)}) = e^{ℓ_{*} / 2} D^{- 1} (\infty) z + O (1)

(7.31)

as z → ∞. On the other hand, we get from (7.16) that there exist integers j, m such that

\tilde{a} (p^{(1)}) - \tilde{a} (p^{(0)}) = ς + ω + B τ + j + B m .

(7.32)

Thus, we get similarly to (7.23) and (7.25) that the function

Θ (z) ≔ \exp \{- 2 π i (m + τ) a (z)\} \frac{θ (a (z) - \tilde{a} (p^{(1)}) - \frac{B + 1}{2})}{θ (a (z) - \tilde{a} (p^{(0)}) - \frac{B + 1}{2})}

(7.33)

is meromorphic in $S \ {α \cup β}$ ⁠, with a simple zero at p⁽¹⁾, a simple pole at p⁽⁰⁾, and otherwise non-vanishing and finite, whose traces on the cycles of the homology basis satisfy

Θ_{+} (s) = Θ_{-} (s) \{\begin{matrix} \exp \{- 2 π i (ω + ς)\}, & s \in α \ {b_{1}}, \\ \exp \{- 2 π i τ\}, & s \in β \ {b_{1}} . \end{matrix})

(7.34)

Combining (7.30), (7.31), and (7.34) yields that the function (FΘ)(z) is rational on $S$ with the divisor p⁽¹⁾ + ∞⁽¹⁾ − p⁽⁰⁾ − ∞⁽⁰⁾. That is, this function is a constant multiple of the reciprocal of (7.13). Recall the definition of S₁ in Theorem 4.2. Observe that

A (\infty) = 1 and B (z) = \frac{S_{1}}{4 i} \frac{1}{z} + O (z^{- 2})

as z → ∞. Then, we get from (7.13) and (7.31) and the symmetry F(z)F(z^*) ≡ 1 that

e^{ℓ_{*}} {= D}^{2} (\infty) \frac{16}{S_{1}^{2}} \frac{Θ (\infty^{(1)})}{Θ (\infty^{(0)})} .

(7.35)

Thus, it only remains to obtain the expression for the last fraction. We get from (7.19) that

a (\infty^{(0)}) - \tilde{a} (p^{(1)}) = \frac{1 + B}{2} + u + B v

for some integers u, v. Hence, we get from (7.7), (7.32), and (7.22) that

Θ (\infty^{(0)}) = \frac{e^{- π i (m + τ) (ς + ω + B τ)} θ (B v)}{θ (ς + ω + B τ + B (m + v))} = \frac{e^{π i (2 v + m - τ) (ς + ω + B τ) + π i B (2 v + m)} θ (0)}{θ (ς + ω + B τ)}

and similarly

\frac{1}{Θ (\infty^{(1)})} = \frac{e^{- π i (m + τ) (ς + ω + B τ)} θ (B (m + v))}{θ (B v - ς - ω - B τ)} = \frac{e^{- π i (2 v + m + τ) (ς + ω + B τ) - π i B (2 v + m)} θ (0)}{θ (- ς - ω - B τ)} .

Since θ(·) is an even function, this finishes the proof of the proposition.

VIII. ASYMPTOTIC ANALYSIS

A. Initial Riemann–Hilbert problem

As agreed before, we omit the dependence on t whenever it does not cause ambiguity. In what follows, it will be convenient to set

I ≔ (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}), σ_{1} ≔ (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}), and σ_{3} ≔ (\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}) .

We are seeking solutions of the following sequence of Riemann–Hilbert problems for 2 × 2 matrix functions (RHP-Y):

Y(z) is analytic in $C \ Γ$ and $\lim_{C \ Γ ∋ z \to \infty} Y (z) z^{- n σ_{3}} = I$ ⁠;
Y(z) has continuous traces on Γ\{a₁, b₁, a₂, b₂} that satisfy
$Y_{+} (s) = Y_{-} (s) (\begin{matrix} 1 & e^{- N_{n} V (s)} \\ 0 & 1 \end{matrix}),$
where, as before, V(z) is given by (1.11) and the sequence {N_n} is such that |n − N_n| ≤ N_* for some N_* > 0.
The connection of RHP-Y to orthogonal polynomials was first demonstrated by Fokas, Its, and Kitaev in Ref. 40 and lies in the following. If the solution of RHP-Y exists, then it is necessarily of the form
$Y (z) = (\begin{matrix} P_{n} (z) & (C P_{n} e^{- N_{n} V}) (z) \\ - \frac{2 π i}{h_{n - 1}} P_{n - 1} (z) & - \frac{2 π i}{h_{n - 1}} (C P_{n - 1} e^{- N_{n} V}) (z) \end{matrix}),$
(8.1)
where P_n(z) = P_n(z; t, N_n) are the polynomial satisfying orthogonality relations (1.10), h_n = h_n(t, N_n) are the constants defined in (1.12), and $C f (z)$ is the Cauchy transform of a function f given on Γ, i.e.,
$(C f) (z) ≔ \frac{1}{2 π i} \int_{Γ} \frac{f (s)}{s - z} d s .$
Below, we show the solvability of RHP-Y for all $n \in N (t, ε)$ large enough following the framework of the steepest descent analysis introduced by Deift and Zhou.⁴¹ The latter lies in a series of transformations that reduce RHP-Y to a problem with jumps asymptotically close to identity.

B. Renormalized Riemann–Hilbert problem

Suppose that Y(z) is a solution of RHP-Y. Put

T (z) ≔ e^{n ℓ_{*} σ_{3} / 2} Y (z) e^{- n (g (z) + ℓ_{*} / 2) σ_{3}},

(8.2)

where the function g(z) is defined by (6.2) and ℓ_* was introduced in (6.3). Then,

T_{+} (s) = T_{-} (s) (\begin{matrix} e^{- n (g_{+} (s) - g_{-} (s))} & e^{n (g_{+} (s) + g_{-} (s) - V (s) + ℓ_{*}) + (n - N_{n}) V (s)} \\ 0 & e^{- n (g_{-} (s) - g_{+} (s))} \end{matrix}),

s ∈ Γ, and therefore, we deduce from (4.7) and (6.6)–(6.8) that T(z) solves RHP-T.

T(z) is analytic in $C \ Γ$ and $\lim_{C \ Γ ∋ z \to \infty} T (z) = I$ ⁠;
T(z) has continuous traces on Γ\{a₁, b₁, a₂, b₂} that satisfy
$T_{+} (s) = T_{-} (s) \{\begin{matrix} (\begin{matrix} 1 & e^{n (2 π i τ + ϕ_{a_{1}} (s)) + (n - N_{n}) V (s)} \\ 0 & 1 \end{matrix}), & s \in Γ (e^{i π} \infty, a_{1}), \\ (\begin{matrix} 1 & e^{n ϕ_{b_{2}} (s) + (n - N_{n}) V (s)} \\ 0 & 1 \end{matrix}), & s \in Γ (b_{2}, e^{π i / 3} \infty), \\ (\begin{matrix} e^{2 π i ω n} & e^{n ϕ_{a_{2}} (s) + (n - N_{n}) V (s)} \\ 0 & e^{- 2 π i ω n} \end{matrix}), & s \in Γ (b_{1}, a_{2}), \end{matrix})$
and
$T_{+} (s) = T_{-} (s) \{\begin{matrix} (\begin{matrix} e^{- n ϕ_{b_{2} +} (s)} & e^{(n - N_{n}) V (s)} \\ 0 & e^{- n ϕ_{b_{2} -} (s)} \end{matrix}), & s \in Γ (a_{2}, b_{2}), \\ (\begin{matrix} e^{- n (ϕ_{b_{2} +} (s) - 2 π i τ)} & e^{2 π i τ n + (n - N_{n}) V (s)} \\ 0 & e^{- n (ϕ_{b_{2} -} (s) - 2 π i τ)} \end{matrix}), & s \in Γ (a_{1}, b_{1}) . \end{matrix})$
Clearly, if RHP-T is solvable and T(z) is the solution, then by inverting (8.2), one obtains a matrix Y(z) that solves RHP-Y.

C. Lens opening

As usual in the steepest descent analysis of matrix Riemann–Hilbert problems for orthogonal polynomials, the next step is based on the identity

\begin{array}{l} (\begin{matrix} e^{- n (ϕ_{b_{2} +} (s) - C)} & e^{n C + (n - N_{n}) V (s)} \\ 0 & e^{- n (ϕ_{b_{2} -} (s) - C)} \end{matrix}) & = (\begin{matrix} 1 & 0 \\ e^{- n ϕ_{b_{2} -} (s) - (n - N_{n}) V (s)} & 1 \end{matrix}) \\ \times (\begin{matrix} 0 & e^{n C + (n - N_{n}) V (s)} \\ - e^{- n C - (n - N_{n}) V (s)} & 0 \end{matrix}) (\begin{matrix} 1 & 0 \\ e^{- n ϕ_{b_{2} +} (s) - (n - N_{n}) V (s)} & 1 \end{matrix}) \end{array}

that follows from (6.6), where C = 2πiτ when s ∈ Γ(a₁, b₁) and C = 0 when s ∈ Γ(a₂, b₂). To carry it out, we shall introduce two additional systems of arcs.

Denote by J_± smooth homotopic deformations of J_t within the region $R e (ϕ_{b_{2}} (z)) > 0$ such that J₊ lies to the left and J₋ to the right of J_t (see Fig. 14) [both connected subarcs of J₊ (respectively, J₋) are oriented from a_i to b_i, i ∈ {1, 2}]. We shall fix the way these arcs emanate from e ∈ {a₁, b₁, a₂, b₂}. In particular, let U_e be given by (6.9) and ${(- ϕ_{e})}^{2 / 3} (z)$ be as in (6.14). Then, we require that

a r g ({(- ϕ_{e})}^{2 / 3} (z)) = \pm ν_{e} (2 π / 3), z \in U_{e} \cap J_{\pm},

(8.3)

where ν_e is defined by (6.13). This requirement always can be fulfilled due to conformality ${(- ϕ_{e})}^{2 / 3} (z)$ in U_e and the choice of the branch in (6.14).

FIG. 14.

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The thick curves represent Γ and thinner black curves represent J_±. The shaded part represents regions where Re(ϕ_e(z)) < 0 (they are the same for each e ∈ {a₁, b₁, a₂, b₂}).

Denote by O_± the open sets delimited by J_± and J_t. Set

S (z) ≔ T (z) \{\begin{matrix} (\begin{matrix} 1 & 0 \\ \mp e^{- n ϕ_{b_{2}} (z) - (n - N_{n}) V (z)} & 1 \end{matrix}), & z \in O_{\pm}, \\ I, & otherwise . \end{matrix})

(8.4)

Then, if T(z) solves RHP-T, S(z) solves RHP-S:

S(z) is analytic in $C \ (Γ \cup J_{+} \cup J_{-})$ and $\lim_{C \ Γ ∋ z \to \infty} S (z) = I$ ⁠;
S(z) has continuous traces on Γ\{a₁, b₁, a₂, b₂} that satisfy RHP-T (b) on Γ(e^πi∞, a₁), Γ(b₁, a₂), and Γ(b₂, e^πi/3∞), as well as
$S_{+} (s) = S_{-} (s) \{\begin{matrix} (\begin{matrix} 0 & e^{(n - N_{n}) V (s)} \\ - e^{- (n - N_{n}) V (s)} & 0 \end{matrix}), & s \in Γ (a_{2}, b_{2}), \\ (\begin{matrix} 0 & e^{2 π i τ n + (n - N_{n}) V (s)} \\ - e^{- 2 π i τ n - (n - N_{n}) V (s)} & 0 \end{matrix}), & s \in Γ (a_{1}, b_{1}), \\ (\begin{matrix} 1 & 0 \\ e^{- n ϕ_{b_{2}} (s) - (n - N_{n}) V (s)} & 1 \end{matrix}), & s \in J_{\pm} . \end{matrix})$
As before, since transformation (8.4) is invertible, a solution of RHP-S yields a solution of RHP-T.

D. Global parametrix

The Riemann–Hilbert problem for the global parametrix is obtained from RHP-S by removing the quantities that are asymptotically zero from the jump matrices in RHP-S (b). The latter can be easily identified with the help of (6.5) and by recalling that the constant τ is real. Thus, we are seeking the solution of RHP-N:

N(z) is analytic in $\bar{C} \ Γ [a_{1}, b_{2}]$ and N(∞) = I;
N(z) has continuous traces on Γ(a₁, b₂)\{b₁, a₂} that satisfy
$N_{+} (s) = N_{-} (s) \{\begin{matrix} (\begin{matrix} 0 & e^{(n - N_{n}) V (s)} \\ - e^{- (n - N_{n}) V (s)} & 0 \end{matrix}), & s \in Γ (a_{2}, b_{2}), \\ (\begin{matrix} 0 & e^{2 π i τ n + (n - N_{n}) V (s)} \\ - e^{- 2 π i τ n - (n - N_{n}) V (s)} & 0 \end{matrix}), & s \in Γ (a_{1}, b_{1}), \\ (\begin{matrix} e^{2 π i ω n} & 0 \\ 0 & e^{- 2 π i ω n} \end{matrix}), & s \in Γ (b_{1}, a_{2}) . \end{matrix})$
We shall solve this problem only for $n \in N (ε) = N (t, ε)$ from Proposition 4.4. In fact, to solve RHP-N, we only need to exclude indices n for which z_n,1 = ∞⁽⁰⁾ (in this case z_n,0 = ∞⁽¹⁾; see Proposition 7.1) as should be clear from (8.5). However, for further analysis, we shall need estimate (7.29), where it is crucial that $n \in N (ε)$ for some ɛ > 0.

Let the functions M_n,k(z) be given by (7.26) and $D (z) = D (z; t)$ be defined by (4.2). With the notation introduced right after (7.23), a solution of RHP-N is given by

N (z) = M^{- 1} (\infty) M (z), M (z) ≔ {(\begin{matrix} M_{n, 1}^{(0)} (z) & M_{n, 1}^{(1)} (z) \\ M_{n, 0}^{(0)} (z) & M_{n, 0}^{(1)} (z) \end{matrix}) D}^{(N_{n} - n) σ_{3}} (z) .

(8.5)

Indeed, RHP-N (a) follows from holomorphy of $D (z)$ and M_n,k(z) discussed in Proposition 4.1 and right after (7.26). Fulfillment of RHP-N (b) can be checked by using (4.3) and (7.27). Observe also that det(N(z)) ≡ 1. Indeed, as the jump matrices in RHP-N (b) have unit determinants, det(N(z)) is holomorphic through Γ(a₁, b₁), Γ(b₁, a₂), and Γ(a₂, b₂). It also has at most square root singularities at {a₁, b₁, a₂, b₂} as explained right after (7.27). Thus, it is holomorphic throughout $\bar{C}$ and, therefore, is a constant. The normalization at infinity implies that this constant is 1.

E. Local parametrices

The jumps discarded in RHP-N are not uniformly close to the identity around each point e ∈ {a₁, b₁, a₂, b₂}. The goal of this section is to solve RHP-S in the disks U_e [see (6.9)] with a certain matching condition on the boundary of the disks. More precisely, we are looking for a matrix functions P_e(z) that solves RHP-P:

P_e(z) has the same analyticity properties as S(z) restricted to U_e [see RHP-S (a)];
P_e(z) satisfies the same jump relations as S(z) restricted to U_e [see RHP-S (b)];
$P_{e} (z) = N (z) (I + O (n^{- 1}))$ holds uniformly on ∂U_e as n → ∞.

Again, we shall solve RHP-P only for $n \in N (ε)$ with $O (\cdot) {= O}_{ε} (\cdot)$ in RHP-P(c).

Let U_e, J_e, and I_e, e ∈ {a₁, b₁, a₂, b₂}, be as in (6.9) and (6.10). Furthermore, let A(ζ) be the Airy matrix.^42,43 That is, it is analytic in $C \ ((- \infty, \infty) \cup L_{-} \cup L_{+})$ ⁠, $L_{\pm} ≔ \{ζ : a r g (ζ) = \pm 2 π / 3\}$ ⁠, and satisfies

A_{+} (s) = A_{-} (s) \{\begin{matrix} (\begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix}), & s \in (- \infty, 0), \\ (\begin{matrix} 1 & 0 \\ 1 & 1 \end{matrix}), & s \in L_{\pm}, \\ (\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}), & s \in (0, \infty), \end{matrix})

where the real line is oriented from −∞ to ∞ and the rays L_± are oriented toward the origin. It is known that A(ζ) has the following asymptotic expansion⁴⁴ at infinity:

A (ζ) e^{\frac{2}{3} ζ^{3 / 2} σ_{3}} \sim \frac{ζ^{- σ_{3} / 4}}{\sqrt{2}} \sum_{k = 0}^{\infty} (\begin{matrix} s_{k} & 0 \\ 0 & t_{k} \end{matrix}) (\begin{matrix} {(- 1)}^{k} & i \\ {(- 1)}^{k} i & 1 \end{matrix}) {(\frac{2}{3} ζ^{3 / 2})}^{- k},

(8.6)

where the expansion holds uniformly in $C \ ((- \infty, \infty) \cup L_{-} \cup L_{+})$ ⁠, and

s_{0} = t_{0} = 1, s_{k} = \frac{Γ (3 k + 1 / 2)}{5 4^{k} k! Γ (k + 1 / 2)}, t_{k} = - \frac{6 k + 1}{6 k - 1} s_{k}, k \geq 1 .

Let us write A_e ≔ A if e ∈ {b₁, b₂} and A_e ≔ σ₃Aσ₃ if e ∈ {a₁, a₂}. It can be easily checked that σ₃Aσ₃ has the same jumps as A only with the reversed orientation of the rays. Moreover, one needs to replace i by −i in (8.6) when describing the behavior of σ₃Aσ₃ at infinity. Let $ζ_{e} (z) ≔ {[- n (3 / 4) ϕ_{e} (z)]}^{2 / 3}$ ⁠, which is conformal in U_e [see (6.14)]. Furthermore, put

J_{e} (z) ≔ e^{(N_{n} - n) V (z) σ_{3} / 2} \{\begin{matrix} I, & e = b_{2}, \\ e^{π i (\pm ω) n σ_{3}}, & e = a_{2}, \\ e^{π i (\pm ω - τ) n σ_{3}}, & e = b_{1}, \\ e^{- π i τ n σ_{3}}, & e = a_{1}, \end{matrix})

where we use ω if z lies to the left of Γ and use −ω if z lies to the right of Γ. Then, it can be readily verified by using (6.5) that

P_{e} (z) ≔ E_{e} (z) A_{e} (ζ_{e} (z)) e^{(2 / 3) ζ_{e}^{3 / 2} (z) σ_{3}} J_{e} (z)

(8.7)

satisfies RHP-P (a) and (b) for any matrix E_e(z) holomorphic in U_e. It follows immediately from (8.6) that RHP-P (c) will be satisfied if

E_{e} (z) ≔ ({N J}_{e}^{- 1}) (z) (\begin{matrix} 1 & - ν_{e} i \\ - ν_{e} i & 1 \end{matrix}) \frac{ζ_{e}^{σ_{3} / 4} (z)}{\sqrt{2}},

(8.8)

provided this matrix function is holomorphic in U_e, where ν_e was defined in (6.13). By using RHP-N (b) and (6.15), one can readily check that E_e(z) is holomorphic in U_e\{e}. Since ζ_e(z) has a simple zero at e, it also follows from (8.5) and the claim after (7.27) that E_e(z) can have at most square root singularity at e and, therefore, is, in fact, holomorphic in the entire disk U_e as needed.

In fact, it follows from (8.6)–(8.8) that

P_{e} (z) \sim N (z) (I + \frac{1}{n} \sum_{k = 0}^{\infty} \frac{P_{e, k} (z)}{n^{k}}),

(8.9)

where the expansion inside the parentheses holds uniformly on ∂U_e and locally uniformly for $t \in O_{two−cut}$ and

P_{e, k - 1} (z) = J_{e}^{- 1} (z) (\begin{matrix} 1 & - ν_{e} i \\ - ν_{e} i & 1 \end{matrix}) (\begin{matrix} s_{k} & 0 \\ 0 & t_{k} \end{matrix}) (\begin{matrix} {(- 1)}^{k} & ν_{e} i \\ ν_{e} {(- 1)}^{k} i & 1 \end{matrix}) J_{e} (z) {(- \frac{ϕ_{e} (z)}{2})}^{- k}, k \geq 1 .

(8.10)

F. RH problem with small jumps

Set $Σ ≔ ([(Γ \ J_{t}) \cup J_{+} \cup J_{-}] \cap D) \cup (\cup_{e} \partial U_{e})$ ⁠, $D ≔ C \ \cup_{e} {\bar{U}}_{e}$ We shall show that for all $n \in N (ε)$ large enough there exists a matrix function R(z) that solves the following Riemann–Hilbert problem (RHP-R):

R(z) is holomorphic in $C \ Σ$ and $\lim_{C \ Γ ∋ z \to \infty} R (z) = I$ ⁠;
R(z) has continuous traces on Σ° (points with the well-defined tangent) that satisfy
$R_{+} (s) = R_{-} (s) \{\begin{cases} P_{e} (s) N^{- 1} (s), & s \in \partial U_{e}, \\ N (s) (\begin{matrix} 1 & 0 \\ e^{- n ϕ_{b_{2}} (s) - (n - N_{n}) V (s)} & 1 \end{matrix}) N^{- 1} (s), & s \in J_{\pm} \cap D, \end{cases})$
where ∂U_e is oriented clockwise, and
$R_{+} (s) = R_{-} (s) \{\begin{matrix} N (s) (\begin{matrix} 1 & e^{n (2 π i τ + ϕ_{a_{1}} (s)) + (n - N_{n}) V (s)} \\ 0 & 1 \end{matrix}) N^{- 1} (s), & s \in Γ (e^{i π} \infty, a_{1}) \cap D, \\ N_{-} (s) (\begin{matrix} e^{2 π i ω n} & e^{n ϕ_{a_{2}} (s) + (n - N_{n}) V (s)} \\ 0 & e^{- 2 π i ω n} \end{matrix}) N_{+}^{- 1} (s), & s \in Γ (b_{1}, a_{2}) \cap D, \\ N (s) (\begin{matrix} 1 & e^{n ϕ_{b_{2}} (s) + (n - N_{n}) V (s)} \\ 0 & 1 \end{matrix}) N^{- 1} (s), & s \in Γ (b_{2}, e^{π i / 3} \infty) \cap D . \end{matrix})$
Observe that RHP-R is a well posed problem as det(N(z)) ≡ 1, as explained after (8.5), and therefore, the matrix is invertible. Recall also that the entries of N(z) and N⁻¹(z) are uniformly bounded on Σ for $n \in N (ε)$ according to (7.28) and (7.29).

To prove solvability of RHP-R, let us show that the jump matrices in RHP-R (b) are close to the identity. To this end, set

Δ (s) ≔ (R

Investigation of the two-cut phase region in the complex cubic ensemble of random matrices

I. INTRODUCTION

II. EQUILIBRIUM MEASURES AND S-PROPERTY

III. STRUCTURE OF Γt

IV. MAIN RESULTS

V. S-CURVES

A. On quadratic differentials

B. Proof of (3.5)

C. Critical graph of ϖt(z)

D. Dependence on t

E. Proof of (3.6)

VI. FUNCTIONS D(z;t) AND Q(z;t)

A. Proof of Proposition 4.1

B. Proof of Proposition 4.2

C. Local analysis at e ∈ {a1, b1, a2, b2}

VII. FUNCTIONS Θn(z; t)

A. Riemann surface

B. Jacobi inversion problem

C. Proof of Proposition 4.5

D. Proof of Proposition 4.6

E. Proof of Proposition 4.4

F. Proof of Proposition 4.3

VIII. ASYMPTOTIC ANALYSIS

A. Initial Riemann–Hilbert problem

B. Renormalized Riemann–Hilbert problem

C. Lens opening

D. Global parametrix

E. Local parametrices

F. RH problem with small jumps

III. STRUCTURE OF Γ_t

C. Critical graph of ϖ_t(z)

VI. FUNCTIONS $D (z; t)$ AND $Q (z; t)$

C. Local analysis at e ∈ {a₁, b₁, a₂, b₂}

VII. FUNCTIONS Θ_n(z; t)