We investigate the phase diagram of the complex cubic unitary ensemble of random matrices with the potential $V(M)=\u221213M3+tM$, 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, $t\u2208C$, is partitioned into two phase regions, $Oone\u2212cut$ and $Otwo\u2212cut$, such that in $Oone\u2212cut$ the equilibrium measure is supported by one Jordan arc (cut) and in $Otwo\u2212cut$ by two cuts. The regions $Oone\u2212cut$ and $Otwo\u2212cut$ 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 $V(M)=\u221213M3+tM$, 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,

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

{see Ref. 4 [formula (1.2.18)]}. The random matrix partition function $ZNRM(t)$ 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

which is well defined (the integrals are convergent) for all values $t\u2208C$.

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 *M*^{2}/2 − *uM*^{3}, where *u* > 0, and the corresponding eigenvalue matrix integral is given by

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

which yields that

where

This implies the relation between the partition functions,

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

for any *u* in the interval 0 ≤ *u* < *u*_{c}, where *u*_{c} ≔ 3^{1/4}18^{−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*,

then the coefficient $f2j(2g)$ is a positive integer that counts the number of 3-valent connected graphs with 2*j* 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 Forrester^{7} (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}

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

and it can be shown that there exists a one-parameter family of solutions {*y*_{α}(*λ*)}, $\alpha \u2208R$, of the Painlevé I differential equation

with $a0=252394$ and $a1=2323\u221254$, 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 2*π*/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 2*π*/5 in the complex plane. We refer to the reader to Ref. 1 (Sec. 2.1) and also the work of Joshi and Kitaev^{16} 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 $\gamma n2$ 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*) = *z*^{n} + ⋯ defined via orthogonality relations

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\u2212\epsilon \u2264nN\u22641+\epsilon $, the recurrence coefficients $\gamma n2$ and *β*_{n} admit asymptotic expansions in powers of $1N2$,

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

where $v\u2208R$ is a scaling variable. Then, as proven in Ref. 1, at *u* = *u*_{c}, the recurrence coefficients $\gamma n2$ and *β*_{n} admit asymptotic expansions in powers of *N*^{−2/5} as *N* → *∞*,

where the functions *p*_{2k}(*v*), $q2k(v\u0303)$ are expressed in terms of the Boutroux tritronquée solution to the Painlevé I equation, $\gamma c2,\beta c$ are known explicitly, and $v\u0303=v+N\u22121/5/2$. 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

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

where the regular and singular factors are given by

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

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

where

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

where *D*_{−1}(*t*, *N*) ≡ 1 and $Dn(t,N)\u2254det\u222b\Gamma zi+je\u2212NV(z;t)dzi,j=0n$ 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

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 *n*th polynomials appearing (1.13) have the prescribed degrees], where

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, $\gamma n2(t,N)=0$, 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 $\gamma N2(t,N)=0$ 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 $\gamma N2(t,N)$ satisfy the Toda equation,

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

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 Lejon^{21} and Kuijlaars and Silva.^{22} Let us start with some definitions.

*Let*

*V*

*be an entire function. The logarithmic energy in the external field*Re

*V*

*of a measure*

*ν*

*in the complex plane is defined as*

*The equilibrium energy of a contour*Γ

*in the external field*Re

*V*

*is equal to*

*where*$M(\Gamma )$

*denotes the space of Borel probability measures on*Γ

*.*

When Re*V*(*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 Re*V*, 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,

where *ℓ* = *ℓ*_{Γ} is a constant, the Lagrange multiplier, and *U*^{μ}(*z*)≔ −*∫*log|*z* − *s*|d*μ*(*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 $\Gamma \u2208T$, where $T$ is the following class of contours:

*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*)

*,*$s\u2208R$

*, for which there exists*

*ɛ*∈ (0,

*π*/6)

*and*

*s*

_{0}> 0

*such that*

*where*arg(

*z*(

*s*)) ∈ [0, 2

*π*)

*.*

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 Re*V*. We make this idea precise in the following definition:

*The support*

*J*

_{Γ}

*has the S-property in the external field*Re

*V*

*if it consists of a finite number of open analytic arcs and their endpoints, and on each arc, it holds that*

*where*$\u2202\u2202n+$

*and*$\u2202\u2202n\u2212$

*are the normal derivatives from the*(+)

*- and*(−)

*-sides of*Γ

*. We shall say that a curve*$\Gamma \u2208T$

*is an S-curve in the field*Re

*V*

*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*)dz^{2} is a maximal regular arc on which

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*)dz^{2}] and it is called *short* if it is incident only with finite critical points. We designate the expression critical (orthogonal) graph of −*Q*(*z*)dz^{2} for the totality of the critical (orthogonal) trajectories −*Q*(*z*)dz^{2}.

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

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

*There exists a contour*$\Gamma t\u2208T$*such that*(2.5)$EV(\Gamma t)=sup\Gamma \u2208TEV(\Gamma ).$*The support**J*_{t}*of the equilibrium measure*$\mu t\u2254\mu \Gamma t$*has the S-property in the external field*Re*V*(*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*(2.6)$Q(z;t)=V\u2032(z;t)2\u2212\u222bd\mu t(s)z\u2212s2,z\u2208C\Jt,$*is a polynomial of degree 4.**The support**J*_{t}*consists of some short critical trajectories of the quadratic differential*−*Q*(*z*;*t*)dz^{2},*and the equation*(2.7)$d\mu t(z)=\u22121\pi iQ+1/2(z;t)dz,z\u2208Jt,$*holds on each such critical trajectory, where*$Q1/2(z;t)=12z2+O(z)$*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

where *e* ∈ *J*_{t} is any and the second equality follows from (2.6) [since the constant $\u2113\Gamma t$ 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*)d*z*^{2} 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 $\Gamma t\Jt\u2282{z:U(z;t)<0}$. 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

[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 2*πk*/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

Furthermore, put $\Omega one\u2212cut$ to be the shaded region on Fig. 2(b) and set

where $\Delta split$ connects −2^{−1/3} and *e*^{πi/3}2^{−1/3}, $\Delta birthb$ extends to infinity in the direction of the angle 7*π*/6, while $\Delta birtha$ extends to infinity in the direction of the angle *π*/6. Let

and set

(see Fig. 3). The function *t*(*x*) is holomorphic in $\Omega one\u2212cut$ with a non-vanishing derivative there. It maps $\Omega one\u2212cut$ onto $Oone\u2212cut$ in a one-to-one fashion. Hence, the inverse map *x*(*t*) exists and is holomorphic.

*Below, we adopt the following convention:* Γ(*z*_{1}, *z*_{2}) *(respectively,* Γ[*z*_{1}, *z*_{2}]*) stands for the trajectory or orthogonal trajectory (respectively, the closure of) of the differential* −*Q*(*z*; *t*)d*z*^{2} *connecting* *z*_{1} *and* *z*_{2}*, oriented from* *z*_{1} *to* *z*_{2}*, and* $\Gamma z,ei\theta \u221e$ *[respectively,* $\Gamma ei\theta \u221e,z$*] 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 $t\u2208O\u0304one\u2212cut$.

*Let*

*μ*

_{t}

*and*

*Q*(

*z*;

*t*)

*be as in Theorem 2.1, and let*$Jt=supp(\mu t)$

*. When*$t\u2208O\u0304one\u2212cut$

*, the polynomial*

*Q*(

*z*;

*t*)

*is of the form*

*with*

*a*(

*t*)

*,*

*b*(

*t*)

*, and*

*c*(

*t*)

*given by*

*where*$x(t)$

*is the branch holomorphic in*$Oone\u2212cut$

*satisfying*$x(0)=e\pi i/3$

*. The set*

*J*

_{t}

*consists of a single arc and the following*:

*If*$t\u2208Oone\u2212cut$*, then**J*_{t}= Γ[*a*,*b*]*and an S-curve*$\Gamma t\u2208T$*can be chosen as*$\Gamma e\pi i\u221e,a\u222aJt\u222a\Gamma b,e\pi i/3\u221e$

*when**t**belongs to the connected component bounded by*$S\u222aCsplit\u222ae2\pi i/3S$*[see**Figs. 4(a)–4(e)**];*$\Gamma e\pi i\u221e,a\u222aJt\u222a\Gamma (b,c)\u222a\Gamma c,e\pi i/3\u221e$

*when**t*∈*S**[see*Fig. 4(f)*];*$\Gamma e\pi i\u221e,c\u222a\Gamma (c,a)\u222aJt\u222a\Gamma b,e\pi i/3\u221e$

*when**t*∈*e*^{2πi/3}*S**;*$\Gamma e\pi i\u221e,a\u222aJt\u222a\Gamma b,e\u2212\pi i/3\u221e\u222a\Gamma e\u2212\pi i/3\u221e,c\u222a\Gamma c,e\pi i/3\u221e$

*when**t**belongs to the connected component bounded by*$S\u222aCbirthb$*[see*Fig. 4(g)*];*$\Gamma e\pi i\u221e,c\u222a\Gamma c,e\u2212\pi i/3\u222a\Gamma e\u2212\pi i/3\u221e,a\u222aJt\u222a\Gamma b,e\pi i/3\u221e$

*when**t**belongs to the connected component bounded by*$e2\pi i/3S\u222aCbirtha$*.*

*If*$t=tcr$*(respectively,*$t=e2\pi i/3tcr$*), then**J*_{t}= Γ[*a*,*b*]*,**c**coincides with**b**(respectively,**a**), and an S-curve*$\Gamma t\u2208T$*can be chosen as in Case I(a) [see*Fig. 5(a)*].**If*$t\u2208Csplit$*, then**J*_{t}= Γ[*a*,*c*] ∪ Γ[*c*,*b*]*and an S-curve*$\Gamma t\u2208T$*can be chosen as in Case I(a) [see*Fig. 5(b)*].**If*$t\u2208Cbirthb$*(respectively,*$t\u2208Cbirtha$*), then**J*_{t}= Γ[*a*,*b*]*and an S-curve*$\Gamma t\u2208T$*can be chosen as in Case I(d) [respectively, Case I(e)] [see*Fig. 5(c)*].*

Now, let $Otwo\u2212cut\u2254C\O\u0304one\u2212cut$. Then, the following theorem holds:

*Let*

*μ*

_{t}

*and*

*Q*(

*z*;

*t*)

*be as in Theorem 2.1,*$Jt=supp(\mu t)$

*. When*$t\u2208Otwo\u2212cut$

*, the polynomial*

*Q*(

*z*;

*t*)

*is of the form*

*with*

*a*

_{1}(

*t*)

*,*

*b*

_{1}(

*t*)

*,*

*a*

_{2}(

*t*)

*, and*

*b*

_{2}(

*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*$t\u2208Otwo\u2212cut$

*; however, at no point of*$Otwo\u2212cut$

*any of the functions*

*a*

_{i}(

*t*),

*b*

_{i}(

*t*)

*is analytic. The S-curve*Γ

_{t}

*can be chosen as*

*where*

*J*

_{t}=

*J*

_{t,1}∪

*J*

_{t,2}

*and*$Jt,i\u2254\Gamma ai(t),bi(t)$

*,*

*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*

*as*

*t*→

*t*

^{*}

*with*$t*\u2208Cbirtha\u222ae2\pi i/3tcr$

*,*$t*\u2208Csplit$

*, and*$t*\u2208Cbirthb\u222atcr$

*, respectively.*

We prove Theorem 3.2 in Sec. V.

## IV. MAIN RESULTS

In this section, we assume that $t\u2208Otwo\u2212cut$ and that *Q*(*z*; *t*) and *J*_{t} = *J*_{t,1} ∪ *J*_{t,2} are as in Theorem 3.2. We also put

*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* $\Gamma b1(t),e\u2212\pi i/3\u221e$ *and* $\Gamma e\u2212\pi i/3\u221e,a2(t)$*, we shall replace them by a smooth Jordan arc, say,* *I*_{t}*, connecting* *b*_{1}(*t*) *and* *a*_{2}(*t*) *such that* *I*_{t}° ≔ *I*_{t}\*E*_{t} *lies entirely within the region* ${U(z;t)<0}$ *while there exist* $s1(t)\u2208\Gamma b1(t),e\u2212\pi i/3\u221e$ *and* $s2(t)\u2208\Gamma e\u2212\pi i/3\u221e,a2(t)$ *for which* $\Gamma b1(t),s1(t)\u222a\Gamma s2(t),a2(t)\u2282It\xb0$ *(see* Fig. 14 *).*

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

*Let constants*

*ς*(

*t*)

*and*

*d*

_{i}(

*t*)

*be given by*

*where, as usual, we use the branch*$Q1/2(z;t)=12z2+O(z)$

*as*

*z*→

*∞*

*. Then, the Szegő function*

*is holomorphic and non-vanishing in*$C\u0304\\Gamma [a1(t),b2(t)]$

*with continuous traces on*

*J*

_{t}° ∪

*I*

_{t}°

*that satisfy*

*Denote by*

*μ*

_{1}(

*t*) ≔

*∫s*d

*μ*

_{t}(

*s*)

*the first moment of the equilibrium measure*

*μ*

_{t}

*. Then,*

*as*

*z*→

*∞*

*. In particular, it holds that*$D(\u221e;t)=exp1\u2212t(d1/d0)(t)3$

*.*

We shall also denote by $D(z;t)\u2254D(z;t)/D(\u221e;t)$ 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

(where we deviate slightly from Convention 1 and denote by Γ[*s*_{1}, *s*_{2}] a subarc of Γ_{t} connecting *s*_{1} to *s*_{2}; for example, Γ(*e*^{πi}*∞*, *a*_{2}] includes *I*_{t} rather than the short trajectories connecting *b*_{1} to *a*_{2}). Observe that $U(z;t)=2Re(Q(z;t))$ as defined in (2.8). This function has the following properties:

*Let constants*

*τ*(

*t*),

*ω*(

*t*)

*be given by*

*These constants are necessarily real [in fact,*

*ω*(

*t*) =

*μ*

_{t}(

*J*

_{t,1})

*; see*

*(2.7)*

*]. The function*$eQ(z;t)$

*is holomorphic in*$C\\Gamma [a1(t),b2(t)]$,

*and there exists a constant*

*ℓ*

_{*}(

*t*)

*such that*

*Moreover,*$Q(z;t)$

*possesses continuous traces on*

*J*

_{t}° ∪

*I*

_{t}°

*that are purely imaginary on*

*J*

_{t}°

*and satisfy*

We prove Proposition 4.2 in Sec. VI B. Observe that it follows from Theorem 3.2 that $|eQ(z;t)|$ 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 $Re(\u2113*(t))=\u2113\Gamma t$ [see (2.2)]. Moreover, the following identity holds:

*Set*

*Let*$\theta (u;B(t))$

*be the Riemann theta function associated with*$B(t)$

*, i.e.,*

*Then, it holds that*

We prove this proposition in Sec. VII F. We also explain right after the statement of Proposition 7.1 further below that *b*_{1}(*t*) − *a*_{1}(*t*) + *b*_{2}(*t*) − *a*_{2}(*t*) ≠ 0. Another auxiliary function we need is given by

for $z\u2208C\u0304\Jt$, where the branches are chosen so that the summands are holomorphic in $C\u0304\Jt$ 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 ${Nn}n\u2208N$, 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.

*Functions*Θ

_{n}(

*z*;

*t*)

*are holomorphic in*$C\u0304\\Gamma [a1(t),b2(t)]$

*with at most one zero there. They have continuous traces on*

*I*

_{t}°

*that satisfy*

*Assume that there exists a constant*

*N*

_{*}

*such that*|

*n*−

*N*

_{n}| ≤

*N*

_{*}

*for all*$n\u2208N$

*. Then, for any*

*δ*> 0

*, there exists a constant*

*C*(

*t*,

*δ*,

*N*

_{*})

*such that*

*that is, including the traces on*Γ[

*a*

_{1}(

*t*),

*b*

_{2}(

*t*)]

*. Given*

*ɛ*> 0

*, let*$N(t,\epsilon )$

*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*

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 $\u03d1n(z;t)DNn\u2212n(z;t)enQ(z;t)$ are functions holomorphic in $C\Jt$.

Proposition 4.4 has substance only if the sets $N(t,\epsilon )$ have infinite cardinality. Recall $B(t)$ from (4.9). It follows from the general theory of Riemann surfaces (see Sec. VII A) that $Im(B(t))>0$. In particular, any $s\u2208C$ can be uniquely written as $x+B(t)y$ for some $x,y\u2208R$.

*Write* $\u03c2(t)=:x(t)+B(t)y(t)$*,* $x(t),y(t)\u2208R$*. It holds that*

*if*$x(t),y(t)\u2208Z$*, 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*$N(t,\epsilon )$*;*^{34}*otherwise, for any natural number**N*_{*}> 0,*there exists a choice of**N*_{n}*such that*|*n*−*N*_{n}| ≤*N*_{*}*and*$N(t,\epsilon )=N$*for all**ɛ**small enough.*

On the other hand, if the sequence ${Nn}n\u2208N$ is fixed, we can claim the following proposition:

*Let*${Nn}n\u2208N$

*be a sequence such that*|

*n*−

*N*

_{n}| ≤

*N*

_{*}

*for some*

*N*

_{*}≥ 0

*. The sequence*$N(t,\epsilon )$

*is infinite for all*

*ɛ*

*small enough unless there exist integers*

*d*> 0,

*k*,

*i*

_{1},

*i*

_{2},

*m*

_{1},

*m*

_{2}

*such that*

*where at least one of the fractions*

*i*

_{1}/

*d*

*,*

*i*

_{2}/

*d*

*is irreducible,*

*d*

*is even,*

*i*

_{1}

*and*

*i*

_{2}

*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*$Nnl+1\u2212Nnl\u2208{0,1}$*(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*$N(t,\epsilon )$*.**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*$N(t,\epsilon )$*.*

*Let*$t\u2208Otwo\u2212cut$

*and*${Nn}n=1\u221e$

*be a sequence such that*|

*n*−

*N*

_{n}| ≤

*N*

_{*}

*for some*

*N*

_{*}

*fixed. Let*

*P*

_{n}(

*z*;

*t*,

*N*)

*be defined by*(1.10) and (1.11)

*and*

*Given*

*ɛ*> 0

*, let*$N(t,\epsilon )$

*be as in Proposition 4.4. Then, for all*$n\u2208N(t,\epsilon )$

*large enough, it holds that*

^{35}

*locally uniformly in*$C\Jt$

*; moreover,*

*locally uniformly on*

*J*

_{t}°

*.*

Recall that each *ϑ*_{n}(*z*; *t*) might have a single zero in $C\Jt$. If these zeros accumulate at a point *z*_{*} along a subsequence of $N(t,\epsilon )$, 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 $C\Jt$. 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 $C\u0304\Jt$) 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 $\u03d1n*(z;t)$. To complicate matters further, our analysis requires another related family of ratios of theta functions, which, in the next theorem, we will denote by $\u03d1\u0303n(z;t)$ [see (7.24) for the rigorous definition of these functions]. The Proof of Proposition 4.4, in fact, shows that $\u03d1n*(\u221e;t)=O\epsilon (1)$ and $\u03d1\u0303n(\u221e;t)=O\epsilon (1)$ for $n\u2208N(t,\epsilon )$.

*Let*$Sk(t)\u2254\u2211i=12bik(t)\u2212aik(t)$

*for*

*k*∈ {1, 2}

*. In the setting of Theorem 4.1, it holds for all*$n\u2208N(t,\epsilon )$

*large enough that*

*where*$D(\u221e;t)$

*is as in Proposition 4.1,*

*ℓ*

_{*}(

*t*)

*as in*

*(4.7)*

*, and*$O\epsilon (\u22c5)$

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

*where we agree that*$\u03d1n*(\u221e;t)ddzlog\u03d1n*(1/z;t)z=0=ddz\u03d1n*(1/z;t)z=0$

*even when*$\u03d1n*(\u221e;t)=0$

*.*

Of course, if there exists a subsequence of $N(t,\epsilon )$ along which the values $\u03d1n*(\u221e;t)$ 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 $\gamma n2(t,Nn)$ 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 $t\u2208Otwo\u2212cut$. In Sec. V C, we show that the critical and critical orthogonal graphs of

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 $C\u0304$ that consists of a finite number of trajectories and orthogonal trajectories of this differential. Then, it holds that

where ord(*z*) is the order of *z* with respect to the considered differential and *θ*(*z*) ∈ [0, 2*π*], *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 2*π*/(*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 *L*_{0}, *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 $z\u21a6\u222bz\u2212\varpi t$ maps end domains conformally onto half planes ${z\u2208C|Re(z)>c}$ for some $c\u2208R$ 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 $z\u21a6\u222bz\u2212\varpi t$ maps strip domains conformally onto vertical strips ${w\u2208C|c1<Re(w)<c2}$ for some $c1,c2\u2208R$ depending on the domain and extends continuously to their boundaries. The number *c*_{2} − *c*_{1} is known as the width of a strip domain and can be calculated in terms of *ϖ*_{t}(*z*) as

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 $t\u2208Otwo\u2212cut$, 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

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

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

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 $\Gamma t\u2208T$ [this class of contours was defined right after (2.3)] such that

[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 $t\u2208Otwo\u2212cut$ 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

*k* ∈ {0, 1, 2}, with all branches being principal. It can be readily verified that *x*_{1}(*t*) is analytic in $C\(e2\pi i/3S\u0304,S\u0304)$ (here, $\u22c5\u0304$ means topological closure) [see (3.2) for the definition of the ray *S*] and

Furthermore, noting that the function *x*(*t*), defined after (3.2), maps $\Omega one\u2212cut$ onto $Oone\u2212cut$, 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 *x*_{1}(*t*). In particular, *x*(*t*) can be analytically continued across $Csplit$, $Cbirtha$, and $Cbirthb$ [see (3.2) and Fig. 7]. In what follows, we consider what happens in the case of each of these continuations.

Continue *x*(*t*) into $Otwo\u2212cut$ by either *x*_{2}(*t*) or *x*_{0}(*t*), that is, analytically across either $Cbirthb$ or $Cbirtha$ (see Fig. 7). The first and the last symmetries in (5.8) then yield that *ϖ*_{t}(*z*) is either equal to

Since the set $Otwo\u2212cut$ is symmetric with respect to the line *L*_{π/3}, its rotation by −2*π*/3 is equal to its reflection across the real axis. Thus, the critical graph *ϖ*_{t}(*z*) when $t\u2208Otwo\u2212cut$ and *x*(*t*) is continued by either *x*_{2}(*t*) or *x*_{0}(*t*) being equal to the reflection across the real axis or the rotation by 4*π*/3 of the critical graph of $\varpi t*(z)$ for some *t*_{*} such that $t\u0304*\u2208Otwo\u2212cut$. 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 *x*_{1}(*t*), that is, analytically across $Csplit$. 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 $t\u2208L\pi /3\u2229Otwo\u2212cut$ [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 $t\u2208Otwo\u2212cut$ in the considered case. Observe that it is enough to show that all the trajectories out of *b* approach infinity.

Recall that the trajectories emanating out of *b* are part of the level set ${U(z;t)=0}$ [see (2.8)]. When $t\u2208L\pi /3\u2229Otwo\u2212cut$, 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 $U(\u2212x;t)\u22600$, the trajectories out of *b* will asymptotically behave as on Fig. 8. When *x*(*t*) is continued into $Otwo\u2212cut$ by *x*_{1}(*t*), its values lie within the gray region on Fig. 9(b) [see also Fig. 2(b)]. The values 2*x*^{3}(*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

where the path of integration lies within the shaded domain on Fig. 9(a). Hence, $U(\u2212x;t)=0$ if and only if 2*x*^{3} belongs to a trajectory of −(1 + 1/*s*)^{3}d*s*^{2} emanating from −1. These trajectories are drawn on Figs. 2 and 9(a) (black lines). Thus, $U(\u2212x;t)\u22600$ in the considered case as claimed.

### 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 $t\u2208Otwo\u2212cut$. Recall the properties of the differential *ϖ*_{t}(*z*) = −*Q*(*z*; *t*)d*z*^{2} 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 *z*_{1}(*t*), *z*_{2}(*t*), *z*_{3}(*t*), *z*_{4}(*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

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, *z*_{4}(*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 *a*_{1}(*t*), *b*_{1}(*t*) and *a*_{2}(*t*), *b*_{2}(*t*). If the other two trajectories out of both *a*_{1}(*t*) and *b*_{1}(*t*) approach infinity, one of these zeros again must belong to the boundary of a strip domain with either *a*_{2}(*t*) or *b*_{2}(*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 *b*_{1}(*t*) and *a*_{2}(*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 *z*_{1}(*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 *z*_{1}(*t*) by short critical trajectories, the unbounded critical trajectories out of *z*_{1}(*t*) delimit a half-plane domain and, in particular, must approach infinity along consecutive critical directions [those are given by the angles (2*k* + 1)*π*/6, *k* ∈ {0, …, 5}; see Sec. V A]. Clearly, the same is true for the unbounded critical trajectories out of *z*_{2}(*t*) and *z*_{3}(*t*) as well as for the unbounded critical trajectories out of *a*_{1}(*t*), *b*_{2}(*t*), and the union of the unbounded critical trajectory out of *b*_{1}(*t*), the short critical trajectory connecting *b*_{1}(*t*) to *a*_{2}(*t*), and the unbounded critical trajectory out of *a*_{2}(*t*) in the case where short critical trajectories form a Jordan arc (see Fig. 10).

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

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 $\mu *\u2208M(\Gamma t)$. Moreover, since *μ*_{*} ≠ *μ*_{t}, $EV(\Gamma *)=EV(\mu *)>EV(\mu t)=EV(\Gamma t)$ [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 *a*_{1}(*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* + i*y*. Consider a determinant of the form

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,

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

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}(*z*_{1}, *z*_{2}, *z*_{3}, *z*_{4}), *j* ∈ {1, 2, 3, 4, 5}, be analytic functions in each variable *z*_{i} = *x*_{i} + i*y*_{i}. We would like to compute the Jacobian of the following system of real-valued functions of *x*_{1}, *y*_{1}, …, *x*_{4}, *y*_{4}:

That is, we are interested in

where $fji\u2254\u2202zifj$. By performing the same row and column operations as for the determinant *D* above, we get that

with the constant in front of the determinant coming from (i/2)^{3} × 2^{4} × (−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

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

where $gji(z1,z2,z3,z4)\u2254(z1\u2212zi)\u22121(fji\u2212fj1)(z1,z2,z3,z4)$, *j* ∈ {4, 5} and *i* ∈ {2, 3, 4}. Hence,

Absolutely analogous computation now implies that

where $hji(z1,z2,z3,z4)\u2254(z2\u2212zi)\u22121(gji\u2212gj2)(z1,z2,z3,z4)$, *j* ∈ {4, 5} and *i* ∈ {3, 4}. The above expression immediately yields that

where $kj(z1,z2,z3,z4)\u2254(z3\u2212z4)\u22121(gj4\u2212gj3)(z1,z2,z3,z4)$, *j* ∈ {4, 5}. Finally, let

be a branch such that $w(z)=z2+O(z)$ as *z* → *∞* with branch cuts *γ*_{12} and *γ*_{34} that are bounded, disjoint, and smooth and where *γ*_{ij} connects *z*_{i} to *z*_{j}. Furthermore, select a smooth arc *γ*_{32} disjoint (except for the endpoints) from the previous two. Set

where we integrate *w*(*z*) on the positive side of *γ*_{12}. Let *O* ⊂ {*z*_{i} ≠ *z*_{j}, *i* ≠ *j*, *i*, *j* ∈ {1, 2, 3, 4}} be a domain such that there exist arcs *γ*_{ij}(*z*_{1}, *z*_{2}, *z*_{3}, *z*_{4}) with the above properties for each (*z*_{1}, *z*_{2}, *z*_{3}, *z*_{4}) ∈ *O*, which, in addition, possess parameterizations that depend continuously on each variable *z*_{1}, *z*_{2}, *z*_{3}, *z*_{4}. Then, the functions *f*_{j}(*z*_{1}, *z*_{2}, *z*_{3}, *z*_{4}), *j* ∈ {4, 5}, are analytic in each variable *z*_{i} for (*z*_{1}, *z*_{2}, *z*_{3}, *z*_{4}) ∈ *O*. Furthermore, it can be readily computed that

where the integrals are taken over *γ*_{12} when *j* = 4 and *γ*_{32} when *j* = 5. Trivially, (5.13) can be rewritten as

Now, consider the Riemann surface $S\u2254z\u2254(z,w):w2=(z\u2212z1)(z\u2212z2)(z\u2212z3)(z\u2212z4)$. Denote by $\pi :S\u2192C\u0304$ the natural projection *π*(** z**) =

*z*and write

*w*(

**) for a rational function on $S$ such that**

*z***= (**

*z**z*,

*w*(

**)). Let**

*z***≔**

*β**π*

^{−1}(

*γ*

_{12}) and

**≔**

*α**π*

^{−1}(

*γ*

_{32}). Orient these cycles so that

Observe that the cycles ** α**,

**form the right pair at the point of their intersection and that $S\{\alpha \u222a\beta}$ is simply connected. Hence, the cycles**

*β***,**

*α***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 d**

*β**z*/

*w*(

**). Hence, we get from (5.15) that**

*z*when (*z*_{1}, *z*_{2}, *z*_{3}, *z*_{4}) ∈ *O*, where the last inequality was shown by Riemann.

Now, let $Q(z;t)=14(z\u2212a1(t))(z\u2212b1(t))(z\u2212a2(t))(z\u2212b2(t))$ be the polynomial from Theorem 2.1. It can be easily deduced from (2.6) that

where the functions *f*_{i}(*z*_{1}, *z*_{2}, *z*_{3}, *z*_{4}), *i* ∈ {1, 2, 3}, are given by (5.12).

Fix $t*\u2208Otwo\u2212cut$ and let *δ*^{*} > 0 be small enough so that all four disks ${|z\u2212zi*|\u2264\delta *}$ are disjoint, where $z1*=a1(t*)$, $z2*=b1(t*)$, $z3*=a2(t*)$, and $z4*=b2(t*)$. Let $O\u2254{(z1,z2,z3,z4):|zi\u2212zi*|<\delta *}$, $si*$ be the point of intersection of ${|z\u2212zi*|=\delta *}$ and $Jt*$, *i* ∈ {1, 2, 3, 4}, and $ui*$ be the point of intersection of ${|z\u2212zi*|=\delta *}$ and Γ(*b*_{1}(*t*^{*}), *a*_{2}(*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 $\gamma 12=[z1,s1*]\u222a\Gamma (s1*,s2*)\u222a[s2*,z2]$, $\gamma 32=[z3,u3*]\u222a\Gamma (u3*,u2*)\u222a[u2*,z2]$, and $\gamma 34=[z3,s3*]\u222a\Gamma (s3*,s4*)\u222a[s4*,z4]$, where [*a*, *b*] is the line segment connecting *a* and *b* in $C$ (see Fig. 11). Clearly, the arcs *γ*_{ij} continuously depend on (*z*_{1}, *z*_{2}, *z*_{3}, *z*_{4}) ∈ *O*. Now, the relations (5.9) can be rewritten as

with *f*_{j}(*z*_{1}, *z*_{2}, *z*_{3}, *z*_{4}), *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, $(a1*(t),b1*(t),a2*(t),b2*(t))$, is such that the real and imaginary parts of $ai*(t),bi*(t)$ 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 (*a*_{1}(*t*^{*}), *b*_{1}(*t*^{*}), *a*_{2}(*t*^{*}), *b*_{2}(*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 $ai*(t*)=ai(t*)$ and $bi*(t*)=bi(t*)$].

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

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 $\u2212U*(z;t)$. 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 $C\{a1(t*),b1(t*),a2(t*),b2(t*)}$ as *t* → *t**. Since the critical graph of $\varpi t*(z)$ is the zero-level set of $U*(z;t)$, it converges to the critical graph of $\varpi t*(z)$ 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 $\varpi t*(z)$ has three short critical trajectories, which, due to uniform convergence, necessarily connect $a1*(t)$ to $b1*(t)$, $b1*(t)$ to $a2*(t)$, and $a2*(t)$ to $b2*(t)$ (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 $\varpi t*(z)$. Moreover, let us now take the branch cut for *Q*^{*}(*z*;*t*)^{1/2}, say, $Jt*$, along the short critical trajectories of $\varpi t*(z)$ connecting $ai*(t)$ to $bi*(t)$. Then, the shading on Fig. 6 corresponds to regions where $U*(z;t)$ is positive (white) and negative (gray).

Define $\Gamma t*$ to be the union of the critical orthogonal trajectory of $\varpi t*(z)$ that connects infinity to $a1*(t)$, its short critical trajectories, and the critical orthogonal trajectory that connects $b2*(t)$ to infinity. Orient it so that the positive direction proceeds from $a1*(t)$ to $b2*(t)$. Let the measures $\mu t*$ be given by (2.7) with *Q*(*z*; *t*) replaced by *Q*^{*}(*z*; *t*) and *J*_{t} replaced by $Jt*$. Clearly, each $\mu t*$ is a positive measure. Moreover, it has a unit mass by the Cauchy theorem and since $Q*(z;t)1/2=(z2\u2212t)/2+1/z+O(1/z2)$ due to (5.17) [see also (5.3)]. Thus, it holds that

as *z* → *∞* and *F*^{*}(*z*; *t*) is holomorphic in $C\u0304\Jt*$. 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 $Jt*$ are bounded. It further follows from the Sokhotski–Plemelj formulas [see Ref. 36 (Sec. 1.4)] that

for $s\u2208Jt*$. Hence, *F*^{*}(*z*; *t*) is an entire function and, therefore, is identically zero. This observation, in particular, yields that

for some constant $\u2113t*$ [see also (2.8)]. Since $U*(z;t)$ can be harmonically continued across $Jt*$ by $\u2212U*(z;t)$, we get that $\mu t*$ satisfies (2.4); that is, $Jt*$ has the *S*-property in the field Re(*V*(*z*; *t*)). Since $\Gamma t*\u2208T$, it follows from the uniqueness part of Theorem 2.1(2) that $\mu t*=\mu t$. In particular, $ai*(t)=ai(t)$ and $bi*(t)=bi(t)$, *i* ∈ {1, 2}.

Since any compact subset of $Otwo\u2212cut$ can be covered by finitely many disks where the above considerations hold, the functions *a*_{i}(*t*), *b*_{i}(*t*) continuously depend on $t\u2208Otwo\u2212cut$, 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 $Otwo\u2212cut$ are the functions *a*_{i}(*t*), *b*_{i}(*t*) analytic in *t*. Suppose, for the sake of contradiction, that one of these functions, say, *b*_{2}(*t*), is analytic in *t* at some point $t0\u2208Otwo\u2212cut$. Then, all the other endpoint functions, *a*_{1}(*t*), *b*_{1}(*t*), and *a*_{2}(*t*), are analytic in *t* at *t*_{0} as well. Indeed, from (5.12) and (5.17), we have that

Hence, *a*_{1}(*t*), *b*_{1}(*t*), *a*_{2}(*t*) are roots of the cubic equation

Since *a*_{1}(*t*), *b*_{1}(*t*), *a*_{2}(*t*) are pairwise different, they analytically depend on the coefficients of the cubic equation. That is, they are analytic functions of *t* and *b*_{2}. This implies that *a*_{1}(*t*), *b*_{1}(*t*), and *a*_{2}(*t*) are analytic in *t* at *t*_{0}. The analyticity of *a*_{1}(*t*), *b*_{1}(*t*), *a*_{2}(*t*), and *b*_{2}(*t*) yields that the integral

is also an analytic function of *t* at *t*_{0}. 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 $Otwo\u2212cut$. 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 *a*_{1}(*t*), *b*_{1}(*t*), *a*_{2}(*t*), and *b*_{2}(*t*) are not analytic functions of *t* at any point in $Otwo\u2212cut$.

### 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 $t*\u2208\u2202Otwo\u2212cut$, 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 $ai*,bi*$ such that *e*(*t*_{m}) → *e*^{*} as *m* → *∞*, where *e* ∈ {*a*_{1}, *b*_{1}, *a*_{2}, *b*_{2}}. Clearly, polynomials *Q*(*z*; *t*_{m}) converge uniformly on compact subsets of $C$ to *Q*^{*}(*z*), the polynomial with zeros $ai*,bi*$ and leading coefficient 1/4. Let *ϖ*^{*}(*z*) ≔ −*Q*^{*}(*z*)d*z*^{2}. Then, repeating the argument after (5.19), we get that the critical graphs of $\varpi tm(z)$ 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 $Rt=2\u22121/2maxi|ai(t)|,|bi(t)|$ 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 $R*\u2254supmRtm$. As all the differentials $\varpi tm(z)$ have structurally “the same” critical graph (see Fig. 6) and all the trajectories of $\varpi tm(z)$ 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 $a1*,b1*,a2*,b2*$ as well. Hence, we cannot simultaneously have that $a1*=b1*$ and $a2*=b2*$. 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 $t\u2208Otwo\u2212cut$ 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 further follows from the choice of the constant *ς*, *d*_{1} in (4.1) that

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

as *z* → *∞*. Moreover, since $Q+1/2(s)=\u2212Q\u22121/2(s)$ for *s* ∈ *J*_{t}, we get the first relation in (4.3). The second relation in (4.3) follows from the Plemelj–Sokhotski formula

### B. Proof of Proposition 4.2

Since the arc *I*_{t} is homologous to the short critical trajectory of −*Q*(*z*)d*z*^{2} connecting *b*_{1} and *a*_{2} and *J*_{t,1} = Γ[*a*_{1}, *b*_{1}] is such a trajectory (see Fig. 6 and Conventions 1 and 2), the constants *τ*, *ω* are indeed real. Let

where we take the principal branch of log(· − *s*) holomorphic outside of $\Gamma e\pi i\u221e,s$ and *μ*_{t} is the equilibrium measure defined in (2.7). It follows directly from definition (6.2) that

where, as usual, we take the branch $Q1/2(z)=12z2+O(z)$, *ℓ*_{*} is a constant such that the equality holds at *b*_{2} [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

holomorphically in $C\\Gamma (e\pi i\u221e,b2]$ when *e* = *b*_{2}, in $C\\Gamma [a1,e\pi i/3\u221e)$ when *e* = *a*_{1}, and in $C\\Gamma (e\pi i\u221e,b1)\u222a\Gamma (a2,e\pi i/3\u221e)$ when *e* ∈ {*b*_{1}, *a*_{2}}. Clearly, $\varphi b2(z)=2Q(z)$. One can readily check that

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

and that

### C. Local analysis at *e* ∈ {*a*_{1}, *b*_{1}, *a*_{2}, *b*_{2}}

Given *e* ∈ {*a*_{1}, *b*_{1}, *a*_{2}, *b*_{2}}, let

where *δ*_{e} ∈ (0, 1] to be adjusted later, and we shall specify the function *ρ*(*t*) at the end of this subsection. Set

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*)d*z*^{2} 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

(see Fig. 6). In fact, the same reasoning shows that (6.11) holds not only on *I*_{e}, but on Γ(*e*^{πi}*∞*, *a*_{1}) when *e* = *a*_{1}, on Γ(*b*_{2}, *e*^{πi/3}*∞*) when *e* = *b*_{2}, and for Re(*ϕ*_{e}(*z*)) on Γ(*b*_{1}, *a*_{2}) when *e* ∈ {*b*_{1}, *a*_{2}} (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

where *J*_{s,e} is the subarc of *J*_{e} with endpoints *e* and *s*,

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 $(\u2212\varphi e)2/3(z)$ in *U*_{e} that is positive on *I*_{e} and satisfies $(\u2212\varphi e)2/3(s)=\u22122\pi \mu t(Js,e)2/3$, *s* ∈ *J*_{e}. Since $(\u2212\varphi e)2/3(z)$ has a simple zero at *e*, it is conformal in *U*_{e} for all radii small enough. Altogether, $(\u2212\varphi e)2/3(z)$ maps *e* into the origin, is conformal in *U*_{e}, and satisfies

Furthermore, if we define $(\u2212\varphi e)1/6(z)$ to be holomorphic in *U*_{e}\*J*_{e} and positive on *I*_{e}, then

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 $(\u2212\varphi e)2/3(z)$ 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 $t\u2208Otwo\u2212cut$ 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

We denote by $\pi :S\u2192C\u0304$ 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***for points on $S$ with natural projections**

*a**z*,

*s*,

*a*.

The function *w*(** z**), defined by

*w*

^{2}(

**) =**

*z**Q*(

*z*), is meromorphic on $S$ with simple zeros at the ramification points

*E*_{t}= {

*a*_{1},

*b*_{1},

*a*_{2},

*b*_{2}} =

*π*

^{−1}(

*E*

_{t}), double poles at the points on top of infinity, and is otherwise non-vanishing and finite. Put

where the domains *D*^{(k)} project onto $C\u0304\Jt$ with labels chosen so that $2w(z)=(\u22121)kz2+O(z)$ as ** z** approaches the point on top of infinity within

*D*

^{(k)}. For $z\u2208C\u0304\Jt$, 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

where ** α** is oriented toward

*b*_{1}within

*D*

^{(0)}and

**is oriented so that**

*β***,**

*α***form the right pair at**

*β*

*b*_{1}(see Figs. 12 and 13). It also will be convenient to put $S\alpha ,\beta \u2254S\{\alpha \u222a\beta}$, which is simply connected.

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

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

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

Given the normalized holomorphic differential, we define

where we restrict ** z** as well as the path of integration to $S\alpha ,\beta $. The function $a(z)$ is a holomorphic function in $S\alpha ,\beta $ with continuous traces on

**,**

*α***(away from the point of their intersection) that satisfy**

*β*by the normalization of $H$ and the definition of $B$. Moreover, it continuously extends to $\u2202S\alpha ,\beta $, the topological boundary of $S\alpha ,\beta $ (the rectangle on Fig. 13). Note that

where $\u2202S\alpha ,\beta $ is oriented so that $S\alpha ,\beta $ remains on the left when $\u2202S\alpha ,\beta $ is traversed in the positive direction. On the other hand, the function $(wa)(z)$ is meromorphic in $S\alpha ,\beta $ with only two singularities, both polar, at *∞*^{(0)} and *∞*^{(1)}. Moreover, since *w*(*z*^{*}) = −*w*(** z**) and $a(z*)=\u2212a(z)$, the residues at those poles coincide. Therefore, it holds that

Recall that $wz(0)=Q1/2(z)$, and therefore, it has expansion (6.1) as *z*^{(0)} → *∞*^{(0)}. This and (7.4) also allow us to deduce that

as *z*^{(0)} → *∞*^{(0)}. Therefore, taking into account (4.1) with the definition of ** α** and the symmetry $a(z*)=\u2212a(z)$, we get that

where the path of integration lies entirely in $S\alpha ,\beta $ and is involution-symmetric.

### B. Jacobi inversion problem

Denote by $Jac(S)\u2254C/{Z+BZ}$ the Jacobi variety of $S$. We shall represent elements of $Jac(S)$ as equivalence classes $[s]={s+j+mB:j,m\u2208Z}$, where $s\u2208C$. One can readily deduce from (7.5) that $a(z)$, defined in (7.4), is essentially a holomorphic bijection from $S$ onto $Jac(S)$. Formally, we define this bijection, known as Abel’s map, by

Given $s\u2208C$, we denote by *z*_{[s]} the unique point in $S$ such that $\u222bb2z[s]H=[s]$.

*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*$z[sn,k(t)]$

*of the Jacobi inversion problem with*

*k*∈ {0, 1}

*. Then, for any subsequence*$N*$,

*the point*

*∞*

^{(0)}

*is a topological limit point of*${zn,1}n\u2208N*$

*if and only if*

*∞*

^{(1)}

*is a topological limit point of*${zn,0}n\u2208N*$

*. 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 *b*_{1} + *b*_{2} + *a*_{1} + *a*_{2} = 0 [see also (5.12) and (5.17)]. Thus, if *b*_{2} + *b*_{1} − *a*_{1} − *a*_{2} = 0 were true, then we would have *b*_{1} = −*b*_{2} and *a*_{1} = −*a*_{2}. In this case, *a*_{1}*b*_{1}*a*_{2} + *a*_{1}*b*_{1}*b*_{2} + *a*_{1}*a*_{2}*b*_{2} + *b*_{1}*a*_{2}*b*_{2} = 0. However, this expression must be equal to −4 according to the above references.

*γ*(

*z*) is holomorphic off

*J*

_{t}= supp(

*μ*

_{t}) and the branch is chosen so that

*γ*(

*∞*) = 1. Furthermore, set

*A*(

*z*) was already defined in (4.12). The functions

*A*(

*z*) and

*B*(

*z*) are holomorphic in $C\u0304\Jt$ and satisfy

*AB*)(

*z*) = 0 can be rewritten as

*γ*

^{4}(

*z*) = 1 and has two solutions, namely,

*∞*and the point

*p*from (7.9). 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}≔

*γ*

^{4}(

*J*

_{t,i}),

*i*∈ {1, 2}, which are unbounded arcs connecting the origin to the point at infinity. Let $L\u2282C\u0304\Jt$ be an arc connecting the point at infinity and

*p*. Then,

*γ*

^{4}(

*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

*γ*

^{4}(

*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

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 $p(0)\u2208S$ 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 $\pi p(k)=p$]. Therefore, Abel’s theorem yields that

while relations (7.9), in particular, imply that

Let *z*_{k} be a topological limit of a subsequence ${zni,k}$. Holomorphy of the differential $H$ implies that

as *i* → *∞*, where the integral from *z*_{k} to $zni,k$ is taken along the path that projects into a segment joining *z*_{k} and $zni,k$. 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

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 $N(\epsilon )=N(t,\epsilon )$ from Proposition 4.4 can be equivalently defined as

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

where *s* is any representative of [*s*]. Furthermore, for each *r*[*s*] > 0, choose *ɛ*_{s} > 0 small enough so that

where we also assume that *ɛ*_{s} is small enough so that always $U\epsilon s\u2282S\alpha ,\beta $. Then,

as otherwise it must hold $0<r[s]\u2264|a(z1)\u2212a(z2)|<(2/3)r[s]$.

Assume that either *x* or *y* is not an integer. Let $zn,1\u2212,zn,1,zn,1+$ 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

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

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

Set $K\u2254{k\u2208Z:r[(1\u2212k)\u03c2+\omega +B\tau ]=0}$. We have already shown that $r[\omega +B\tau ]>0$. It also holds that $r[\u03c2+\omega +B\tau ]>0$ since $[\u03c2+\omega +B\tau ]\u2260[0]$ by (7.7) and the unique solvability of the Jacobi inversion problem. The following lemma easily follows from (7.17).

*Assume that* |*n* − *N*_{n}| ≤ *N*_{*}, *and let* $\epsilon k\u2254\epsilon (1\u2212k)\u03c2+\omega +B\tau $*.*

*If*$K=\u2205$*, then at least one of the integers**n*,*n*+ 1*belongs to*$N(\epsilon )$*for all*$\epsilon \u2264min|k|\u22642N*+1\epsilon k$*.**If*$K={k\u2032}$*, there exists an infinite subsequence*{*n*_{l}}*such that*$Nnl+1\u2212Nnl\u2260k\u2032$,*and therefore, at least one of the integers**n*_{l},*n*_{l}+ 1*belongs to*$N(\epsilon )$*for all*$\epsilon \u2264min|k|\u22642N*+1,k\u2260k\u2032\epsilon k$*.**If there exists an infinite subsequence*{*n*_{l}}*such that*$Nnl+1\u2212Nnl\u2208{0,1}$*, then at least one of the integers**n*_{l},*n*_{l}+ 1*belongs to*$N(\epsilon )$*for all**ɛ*≤ min{*ɛ*_{0},*ɛ*_{1}}*.*

Inclusion $k\u2208K$ means that $[(1\u2212k)\u03c2+\omega +B\tau ]=[0]$. Hence, both triples *ω*, *x*, 1 and *τ*, *y*, 1 are rationally dependent, $\u03c2=x+By$. Assume that $k\u2032,k\u2032\u2032\u2208K$, *k*′ ≠ *k*″. Then, it follows from (7.18) that $[\omega +B\tau ]=[(k\u2032\u22121)\u03c2]$ 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 $K=k\u2032+dZ$. 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:

*If one of the triples* *ω*, *x*, 1 *or* *τ*, *y*, 1 *is rationally independent, then* $K=\u2205$*. 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* $K=\u2205$ *or* $K={k\u2032}$*.*

Assume now that all three relations of (4.13) take place. That is, $[\omega +B\tau ]=[(k\u22121)\u03c2]$ and [*dς*] = [0] for some integers *k*, *d*. It follows from (7.14) that

for some $j,m\u2208Z$, 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

i.e., both *j*, *m* must be odd. Indeed, the first equality follows from the symmetry $a(z*)=\u2212a(z)$. Next, observe that the function $(\u22121)k\gamma \u22122z(k)\u22121$ is meromorphic on $S$ with zeros at *∞*^{(0)} and *p*^{(0)} and poles at *b*_{1} and *b*_{2}, 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 *b*_{1} in *D*^{(0)}. It is clearly homologous to ** α** +

**. 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)**

*β**a posteriori*holds with the bounds of integration in the left most integral of (7.19) in flipped. Therefore, adding $\u222b\u221e(0)b2H$ to both sides of (7.9) gives us

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

Clearly, ${zn,1}n\u2208N\u2286{z\rho}\rho =0d\u22121$. Thus, it only remains to investigate when *z*_{ρ} = *∞*^{(0)} or, equivalently, when $[(1+B)/2+\rho \u03c2]=[0]$. From the representation of *ς* in (4.13), it must simultaneously hold that *ρ* = *d*(2*l*_{j} − 1)/(2*i*_{j}), *j* ∈ {1, 2}, for some $l1,l2\u2208Z$. 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} = 2*l*_{j} − 1 must be odd), and the former is ruled out since $[(1+B)/2]\u2260[0]$ (or equivalently, since 0 ≠ 2*l*_{j} − 1).

### 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 $a\u0303(z)$ of $a(z)$ to the whole surface $S$ defined by setting $a\u0303(s)\u2254a+(s)$ for ** s** ∈

**and**

*α***∈**

*s***\{**

*β*

*b*_{1}}. Given such an extension and (7.9), there exist unique integers

*j*

_{n,k},

*m*

_{n,k}such that

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

It is also known that *θ*(*u*) vanishes only at the points of the lattice $B+12$. Let

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

where *F*^{(i)}(*z*), *i* ∈ {0, 1}, stands for the pull-back under *π*(** z**) of a function

*F*(

**) from**

*z**D*

^{(i)}into $C\u0304\Jt$.

The functions Θ_{n,k}(** z**) are meromorphic on $S\alpha ,\beta $ 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***away from**

*β*

*b*_{1}that satisfy

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

These functions are holomorphic on $S\{\alpha \u222a\beta \u222a\Delta}$ since the pole of Θ_{n,k}(** z**) is canceled by the zero of

*B*(

*z*). Each function

*M*

_{n,k}(

**) has exactly two zeros, namely,**

*z*

*z*_{n,k}and

*∞*

^{(k)}. It follows from (7.12) and (7.25) that

It further follows from (7.10) and (7.11) that |*M*_{n,k}(** z**)| ∼ |

*z*−

*e*|

^{−1/4}as

**→**

*z***∈**

*e***unless**

*E*

*z*_{n,k}coincides with

**in which case the exponent becomes 1/4. Assume now that there exists**

*e**N*

_{*}≥ 0 such that |

*n*−

*N*

_{n}| ≤

*N*

_{*}for all $n\u2208N$. Then, for each

*δ*> 0 there exists

*C*(

*δ*,

*N*

_{*}) independent of

*n*such that

Indeed, observe that {log|*M*_{n,k}(** z**)|} is a family of subharmonic functions in

*O*

_{δ}\

**[the jump of**

*α**M*

_{n,k}(

**) is unimodular on**

*z***]. By the maximum principle for subharmonic functions, log|**

*β**M*

_{n,k}(

**)| reaches its maximum on**

*z**∂*(

*O*

_{δ}\

**), where the maximum is clearly finite. Since the sequence {**

*α**n*−

*N*

_{n}} is bounded by assumption and the range of $a\u0303(z)$ 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 $Im(B)>0$. Thus, any limit point of {log|

*M*

_{n,k}(

**)|} is obtained by taking simultaneous limit points of {**

*z**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}\

**, which proves (7.28). Finally, it holds that**

*α*for some constant *c*_{ɛ} > 0 by a similar compactness argument combined with the definition of $N(\epsilon )$ 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 $S\{\alpha \u222a\beta}$ by setting $Fz(1)\u22541/Fz(0)$ and

where one needs to recall that *I*_{t} is oriented toward *a*_{2}, while $\alpha \u2229S(0)$ is oriented toward *b*_{1}. Moreover, it follows from (4.7) that

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

is meromorphic in $S\{\alpha \u222a\beta}$, 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

Combining (7.30), (7.31), and (7.34) yields that the function (*F*Θ)(** 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

*S*

_{1}in Theorem 4.2. Observe that

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

and similarly

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

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\\Gamma $ and $limC\\Gamma \u220bz\u2192\u221eY(z)z\u2212n\sigma 3=I$;(*Y**z*) has continuous traces on Γ\{*a*_{1},*b*_{1},*a*_{2},*b*_{2}} that satisfywhere, as before,$Y+(s)=Y\u2212(s)1e\u2212NnV(s)01,$*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-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*where(8.1)$Y(z)=Pn(z)CPne\u2212NnV(z)\u22122\pi ihn\u22121Pn\u22121(z)\u22122\pi ihn\u22121CPn\u22121e\u2212NnV(z),$*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 $Cf(z)$ is the Cauchy transform of a function*f*given on Γ, i.e.,$(Cf)(z)\u225412\pi i\u222b\Gamma f(s)s\u2212zds.$Below, we show the solvability of RHP-

for all $n\u2208N(t,\epsilon )$ large enough following the framework of the steepest descent analysis introduced by Deift and Zhou.*Y*^{41}The latter lies in a series of transformations that reduce RHP-to a problem with jumps asymptotically close to identity.*Y*

### B. Renormalized Riemann–Hilbert problem

Suppose that ** Y**(

*z*) is a solution of RHP-

**. Put**

*Y*(*T**z*) is analytic in $C\\Gamma $ and $limC\\Gamma \u220bz\u2192\u221eT(z)=I$;(*T**z*) has continuous traces on Γ\{*a*_{1},*b*_{1},*a*_{2},*b*_{2}} that satisfyand$T+(s)=T\u2212(s)1en(2\pi i\tau +\varphi a1(s))+(n\u2212Nn)V(s)01,s\u2208\Gamma (ei\pi \u221e,a1),1en\varphi b2(s)+(n\u2212Nn)V(s)01,s\u2208\Gamma (b2,e\pi i/3\u221e),e2\pi i\omega nen\varphi a2(s)+(n\u2212Nn)V(s)0e\u22122\pi i\omega n,s\u2208\Gamma (b1,a2),$$T+(s)=T\u2212(s)e\u2212n\varphi b2+(s)e(n\u2212Nn)V(s)0e\u2212n\varphi b2\u2212(s),s\u2208\Gamma (a2,b2),e\u2212n(\varphi b2+(s)\u22122\pi i\tau )e2\pi i\tau n+(n\u2212Nn)V(s)0e\u2212n(\varphi b2\u2212(s)\u22122\pi i\tau ),s\u2208\Gamma (a1,b1).$Clearly, if RHP-

is solvable and*T*(*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

that follows from (6.6), where *C* = 2*π*i*τ* when *s* ∈ Γ(*a*_{1}, *b*_{1}) and *C* = 0 when *s* ∈ Γ(*a*_{2}, *b*_{2}). To carry it out, we shall introduce two additional systems of arcs.

Denote by *J*_{±} smooth homotopic deformations of *J*_{t} within the region $Re(\varphi b2(z))>0$ 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* ∈ {*a*_{1}, *b*_{1}, *a*_{2}, *b*_{2}}. In particular, let *U*_{e} be given by (6.9) and $(\u2212\varphi e)2/3(z)$ be as in (6.14). Then, we require that

where *ν*_{e} is defined by (6.13). This requirement always can be fulfilled due to conformality $(\u2212\varphi e)2/3(z)$ in *U*_{e} and the choice of the branch in (6.14).

Denote by *O*_{±} the open sets delimited by *J*_{±} and *J*_{t}. Set

Then, if ** T**(

*z*) solves RHP-

**,**

*T***(**

*S**z*) solves RHP-

**:**

*S*(*S**z*) is analytic in $C\(\Gamma \u222aJ+\u222aJ\u2212)$ and $limC\\Gamma \u220bz\u2192\u221eS(z)=I$;(*S**z*) has continuous traces on Γ\{*a*_{1},*b*_{1},*a*_{2},*b*_{2}} that satisfy RHP-(b) on Γ(*T**e*^{πi}*∞*,*a*_{1}), Γ(*b*_{1},*a*_{2}), and Γ(*b*_{2},*e*^{πi/3}*∞*), as well as$S+(s)=S\u2212(s)0e(n\u2212Nn)V(s)\u2212e\u2212(n\u2212Nn)V(s)0,s\u2208\Gamma (a2,b2),0e2\pi i\tau n+(n\u2212Nn)V(s)\u2212e\u22122\pi i\tau n\u2212(n\u2212Nn)V(s)0,s\u2208\Gamma (a1,b1),10e\u2212n\varphi b2(s)\u2212(n\u2212Nn)V(s)1,s\u2208J\xb1.$As before, since transformation (8.4) is invertible, a solution of RHP-

yields a solution of RHP-*S*.*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-

**(b). The latter can be easily identified with the help of (6.5) and by recalling that the constant**

*S**τ*is real. Thus, we are seeking the solution of RHP-

**:**

*N*(*N**z*) is analytic in $C\u0304\\Gamma [a1,b2]$ and(*N**∞*) =;*I*(*N**z*) has continuous traces on Γ(*a*_{1},*b*_{2})\{*b*_{1},*a*_{2}} that satisfy$N+(s)=N\u2212(s)0e(n\u2212Nn)V(s)\u2212e\u2212(n\u2212Nn)V(s)0,s\u2208\Gamma (a2,b2),0e2\pi i\tau n+(n\u2212Nn)V(s)\u2212e\u22122\pi i\tau n\u2212(n\u2212Nn)V(s)0,s\u2208\Gamma (a1,b1),e2\pi i\omega n00e\u22122\pi i\omega n,s\u2208\Gamma (b1,a2).$We shall solve this problem only for $n\u2208N(\epsilon )=N(t,\epsilon )$ from Proposition 4.4. In fact, to solve RHP-

, we only need to exclude indices*N**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 $n\u2208N(\epsilon )$ 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-

**is given by**

*N*Indeed, RHP-** N** (a) follows from holomorphy of $D(z)$ and

*M*

_{n,k}(

**) discussed in Proposition 4.1 and right after (7.26). Fulfillment of RHP-**

*z***(b) can be checked by using (4.3) and (7.27). Observe also that det(**

*N***(**

*N**z*)) ≡ 1. Indeed, as the jump matrices in RHP-

**(b) have unit determinants, det(**

*N***(**

*N**z*)) is holomorphic through Γ(

*a*

_{1},

*b*

_{1}), Γ(

*b*

_{1},

*a*

_{2}), and Γ(

*a*

_{2},

*b*

_{2}). It also has at most square root singularities at {

*a*

_{1},

*b*

_{1},

*a*

_{2},

*b*

_{2}} as explained right after (7.27). Thus, it is holomorphic throughout $C\u0304$ 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*∈ {

*a*

_{1},

*b*

_{1},

*a*

_{2},

*b*

_{2}}. The goal of this section is to solve RHP-

**in the disks**

*S**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-(a)];*S**P*_{e}(*z*) satisfies the same jump relations as(*S**z*) restricted to*U*_{e}[see RHP-(b)];*S*$Pe(z)=N(z)I+O(n\u22121)$ holds uniformly on

*∂U*_{e}as*n*→*∞*.

Again, we shall solve RHP-** P** only for $n\u2208N(\epsilon )$ with $O(\u22c5)=O\epsilon (\u22c5)$ in RHP-

**(c).**

*P*Let *U*_{e}, *J*_{e}, and *I*_{e}, *e* ∈ {*a*_{1}, *b*_{1}, *a*_{2}, *b*_{2}}, 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\(\u2212\u221e,\u221e)\u222aL\u2212\u222aL+$, $L\xb1\u2254\zeta :arg(\zeta )=\xb12\pi /3$, and satisfies

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 expansion

^{44}at infinity:

where the expansion holds uniformly in $C\(\u2212\u221e,\u221e)\u222aL\u2212\u222aL+$, and

Let us write *A*_{e} ≔ ** A** if

*e*∈ {

*b*

_{1},

*b*

_{2}} and

*A*_{e}≔

*σ*

_{3}

*A**σ*

_{3}if

*e*∈ {

*a*

_{1},

*a*

_{2}}. It can be easily checked that

*σ*

_{3}

*A**σ*

_{3}has the same jumps as

**only with the reversed orientation of the rays. Moreover, one needs to replace i by −i in (8.6) when describing the behavior of**

*A**σ*

_{3}

*A**σ*

_{3}at infinity. Let $\zeta e(z)\u2254\u2212n(3/4)\varphi e(z)2/3$, which is conformal in

*U*

_{e}[see (6.14)]. Furthermore, put

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

satisfies RHP-** P** (a) and (b) for any matrix

*E*_{e}(

*z*) holomorphic in

*U*

_{e}. It follows immediately from (8.6) that RHP-

**(c) will be satisfied if**

*P*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.

where the expansion inside the parentheses holds uniformly on *∂U*_{e} and locally uniformly for $t\u2208Otwo\u2212cut$ and

### F. RH problem with small jumps

Set $\Sigma \u2254(\Gamma \Jt)\u222aJ+\u222aJ\u2212\u2229D\u222a\u222ae\u2202Ue$, $D\u2254C\\u222aeU\u0304e$ We shall show that for all $n\u2208N(\epsilon )$ large enough there exists a matrix function ** R**(

*z*) that solves the following Riemann–Hilbert problem (RHP-

**):**

*R*(*R**z*) is holomorphic in $C\\Sigma $ and $limC\\Gamma \u220bz\u2192\u221eR(z)=I$;(*R**z*) has continuous traces on Σ° (points with the well-defined tangent) that satisfywhere$R+(s)=R\u2212(s)Pe(s)N\u22121(s),s\u2208\u2202Ue,N(s)10e\u2212n\varphi b2(s)\u2212(n\u2212Nn)V(s)1N\u22121(s),s\u2208J\xb1\u2229D,$*∂U*_{e}is oriented clockwise, and$R+(s)=R\u2212(s)N(s)1en(2\pi i\tau +\varphi a1(s))+(n\u2212Nn)V(s)01N\u22121(s),s\u2208\Gamma (ei\pi \u221e,a1)\u2229D,N\u2212(s)e2\pi i\omega nen\varphi a2(s)+(n\u2212Nn)V(s)0e\u22122\pi i\omega nN+\u22121(s),s\u2208\Gamma (b1,a2)\u2229D,N(s)1en\varphi b2(s)+(n\u2212Nn)V(s)01N\u22121(s),s\u2208\Gamma (b2,e\pi i/3\u221e)\u2229D.$Observe that RHP-

is a well posed problem as det(*R*(*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 $n\u2208N(\epsilon )$ according to (7.28) and (7.29).

To prove solvability of RHP-** R**, let us show that the jump matrices in RHP-

**(b) are close to the identity. To this end, set**

*R*Since the entries of ** N**(

*z*) are uniformly bounded on each

*∂U*

_{e}with respect to $n\u2208N(\epsilon )$, it holds by RHP-

**(c) and (8.9) that**

*P*where this pseudo-expansion^{45} is valid uniformly on *∂U*_{e}. Thus, it holds that