Fluctuations in the number of nodal domains

Nazarov, Fedor; Sodin, Mikhail

doi:10.1063/5.0018588

We show that the variance of the number of connected components of the zero set of the two-dimensional Gaussian ensemble of random spherical harmonics of degree n grows as a positive power of n. The proof uses no special properties of spherical harmonics and works for any sufficiently regular ensemble of Gaussian random functions on the two-dimensional sphere with distribution invariant with respect to isometries of the sphere. Our argument connects the fluctuations in the number of nodal lines with those in a random loop ensemble on planar graphs of degree four, which can be viewed as a step toward justification of the Bogomolny–Schmit heuristics.

I. INTRODUCTION

Let (f_n) be the ensemble of random Gaussian spherical harmonics of degree n on the two-dimensional sphere, and let N(f_n) be the number of connected components of the zero set {f_n = 0}. It is known that

E [N (f_{n})] = (c + o (1)) n^{2}, n \to \infty,

with a positive numerical constant c and that the random variable N(f_n) exponentially concentrates around its mean.⁴ A beautiful Bogomolny–Schmit heuristics² suggests that, for any ɛ > 0 and n large enough,

n^{2 - ε} < Var [N (f_{n})] < n^{2 + ε} .

However, the rigorous bounds we are aware of are much weaker,

n^{σ} ≲ Var [N (f_{n})] ≲ n^{4 - σ},

with some σ > 0. The upper bound with $σ = \frac{2}{15}$ follows from the exponential concentration of N(f_n) around its mean (see Remark 1.2 of Ref. 4). The purpose of this paper is to prove the lower bound. The proof we give uses no special properties of spherical harmonics and shows that this lower bound holds for any smooth non-degenerate ensemble of Gaussian random functions on the two-dimensional sphere $S^{2}$ with a distribution invariant with respect to isometries of the sphere and correlations decaying at least as a positive power of the appropriately scaled distance on $S^{2}$ ⁠.

It is worth mentioning that in a recent study,¹ Beliaev, McAuley, and Muirhead found non-trivial lower bounds for fluctuations of the number of connected components in the disk of radius R ≫ 1 of the level sets {F = ℓ} and of the excursion set {F ⩾ ℓ} of the random plane wave (which is a scaling limit of the ensemble of spherical harmonics) for non-zero levels ℓ ≠ 0. It is expected that in their case, the fluctuations are much larger than the ones we study. The techniques used in their work are quite different.

II. THE SETUP AND THE MAIN RESULT

Let (f_L) be an ensemble of Gaussian random functions on the two-dimensional sphere. It is convenient to assume that the function f_L is defined on the sphere $S^{2} (L) = {x \in R^{3} : | x | = L}$ of large radius L and is normalized by $E [f_{L} {(x)}^{2}] = 1$ for all $x \in S^{2} (L)$ ⁠. We always assume that the distribution of f_L is invariant with respect to the isometries of the sphere. Then, the covariance kernel of f_L has the form

K_{L} (x, y) = E [f_{L} (x) f_{L} (y)] = k_{L} (d_{L} (x, y)), x, y \in S^{2} (L),

where d_L is the spherical distance on $S^{2} (L)$ ⁠. We call such an ensemble (f_L) regular if the following two conditions hold:

$C^{3 +}$ -smoothness: $K_{L} \in C^{3 + ν, 3 + ν} (S^{2} (L))$ with estimates uniform in L and with some ν > 0.
Power decay of correlations: $K_{L} (x, y) ≲ {(1 + d_{L} (x, y))}^{- γ}$ ⁠, $x, y \in S^{2} (L)$ ⁠, with some γ > 0 and with the implicit constant independent of L.

Condition (1) yields that almost surely, $f_{L} \in C^{3} (S^{2} (L))$ with estimates uniform in L. We also note that condition (2) is equivalent to the estimate |k_L(d)| ≲ (1 + d)^−γ for 0 ⩽ d ⩽ πL (with the implicit constant independent of L).

By Z(f_L), we denote the random zero set of f_L, which is, almost surely, a collection of disjoint simple smooth closed random curves (“loops”) on $S^{2} (L)$ ⁠. By N(f_L), we denote the number of these loops.

Theorem.

Let (f_L) be a regular Gaussian ensemble. Then, there exists σ > 0 such that, for L ⩾ L₀,

Var [N (f_{L})] ⩾ L^{σ} .

There are many natural regular Gaussian ensembles, but a nuisance is that the spherical harmonics ensemble is not among them. Spherical harmonics are symmetric with respect to the center of the sphere, so their values at the antipodal points on the sphere coincide up to the sign. The correlations for this ensemble still satisfy condition (2) but only in the range 0 ⩽ d ⩽ (π − ɛ)L with any ɛ > 0. For this reason, our theorem cannot be applied to this ensemble directly. Luckily, the case of the spherical harmonics requires only minor modifications in the proof of the theorem, which we will outline in the last section of this work. Essentially, we just need an analogue of our theorem for the projective plane $R P^{2}$ instead of the sphere.

III. MAIN STEPS IN THE PROOF

Heuristically, the fluctuations in the topology of the zero set are caused by fluctuations in the signs of the critical values. To exploit this heuristics, we fix a random function f_L and slightly perturb it by a multiple of its independent copy g_L, i.e., consider the random function

{\tilde{f}}_{L} = \sqrt{1 - α^{' 2}} f_{L} + α^{'} g_{L}, 0 < α^{'} ≪ 1,

which has the same distribution as f_L. Let α be another small parameter, which is significantly bigger than α′, α′ ≪ α ≪ 1, and let

Cr (α) = \{p \in S^{2} (L) : \nabla f_{L} (p) = 0, | f_{L} (p) | ⩽ α\} .

Then, as we will see, with high probability, given f_L, the change in the topology of the zero set $Z ({\tilde{f}}_{L})$ is determined by the signs of ${\tilde{f}}_{L}$ at Cr(α), that is, by the collection of the random values $\{g_{L} (p) : p \in Cr (α)\}$ ⁠. To make the correlations between these random values negligible, the set Cr(α) should be well-separated on the sphere $S^{2} (L)$ ⁠. At the same time, the set Cr(α) has to be relatively large; otherwise, the impact of fluctuations in signs of ${\tilde{f}}_{L}$ on the number $N ({\tilde{f}}_{L})$ will be negligible. In Lemma 17, we will show that

there exist positive ɛ₀, c, and C such that, given 0 < ɛ ⩽ ɛ₀ and L ⩾ L₀(ɛ), for L^−2+ɛ ⩽ α ⩽ L^−2+2ɛ, with probability very close to 1, the set Cr(α) is L^1−Cɛ-separated and |Cr(α)| ⩾ L^cɛ.

The proof of this lemma given in Secs. VII–IX is the longest and probably the most delicate part of our work.

To understand how the signs of ${\tilde{f}}_{L}$ at Cr(α) affect the topology of the zero set $Z ({\tilde{f}}_{L})$ ⁠, we develop in Sec. VI a little caricature of the quantitative Morse theory. This caricature is non-random—its applicability to the random function f_L relies on the fact that with high probability, the Hessian ∇²f_L cannot degenerate at the points where the function f_L and its gradient ∇f_L are simultaneously small. We show that if the parameter α′ is small enough, then with high probability, the topology of $Z ({\tilde{f}}_{L})$ depends only on the signs of the eigenvalues of the Hessian ∇²f_L(p) and the signs of ${\tilde{f}}_{L} (p)$ ⁠, p ∈ Cr(α). We will describe how these signs determine the structure of the zero set $Z ({\tilde{f}}_{L})$ in small neighborhoods of the points p ∈ Cr(α). Outside these neighborhoods, the zero lines of $Z ({\tilde{f}}_{L})$ stay close to the ones of Z(f_L).

First, we consider the critical points p ∈ Cr(α) for which both eigenvalues of the Hessian ∇²f_L(p) have the same sign, i.e., the points that are local extrema of f_L. In this case, we show that there exists a disk D(p, δ) centered at p of a small radius δ such that with high probability, $Z ({\tilde{f}}_{L}) \cap D (p, δ)$ either consists of a simple loop encircling the point p when the sign of ${\tilde{f}}_{L} (p)$ is opposite to that of the eigenvalues of ∇²f_L(p) or is empty when these signs coincide. We call such connected components of $Z ({\tilde{f}}_{L})$ blinking circles.

Now, we turn to the case when the eigenvalues of the Hessian ∇²f_L(p) have opposite signs, i.e., to the saddle points of f_L. In this case, the situation is more intricate. We define a degree four graph G(f_L) embedded in $S^{2} (L)$ ⁠. Its vertices are small neighborhoods J(p, δ) of saddle points p ∈ Cr(α). The edges are arcs in the set Z(f_L) that connect these neighborhoods. We will show that with high probability, the collection of signs ${sgn ({\tilde{f}}_{L} (p)) : p \in Cr (α)}$ determines how the graph G(f_L) is turned into a collection of loops in $Z ({\tilde{f}}_{L})$ ⁠, which we will call the Bogomolny–Schmit loops. Figure 1 illustrates how the sign of ${\tilde{f}}_{L}$ at the saddle point p ∈ Cr(α) determines the structure of the zero set $Z ({\tilde{f}}_{L})$ in a small neighborhood of the saddle point p ∈ Cr(α).

FIG. 1.

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The sign of ${\tilde{f}}_{L}$ determines the structure of the zero set $Z ({\tilde{f}}_{L})$ near the saddle point p.

We see that in both cases, the fluctuations in the number of connected components of $Z ({\tilde{f}}_{L})$ are caused by fluctuations in the signs,

sgn ({\tilde{f}}_{L} (p)) = sgn (\sqrt{1 - α^{' 2}} f_{L} (p) + α^{'} g_{L} (p)), p \in Cr (α) .

We show that, since the points of the set Cr(α) are well separated and the covariance kernel k_L(d) decays at least as a power of d, with probability very close to one, we can replace the values g_L(p), p ∈ Cr(α), by a collection of independent standard Gaussian random variables.

Thus, conditioning on f_L, we may assume that the values of ${\tilde{f}}_{L}$ at Cr(α) are independent normal random variables (not necessarily mean zero). To conclude, we apply Lemma 21 on the variance of the number of loops generated by percolation-like processes on planar graphs of degree 4.

Note that this chain of arguments can be viewed as the first, although very modest, step toward justification of the Bogomolny–Schmit heuristics.

IV. NOTATION

Throughout the paper, we will be using the following notation:

L is a large parameter that tends to +∞. We always assume that L ⩾ 1.
$S^{2} (L)$ denotes the sphere in $R^{3}$ centered at the origin and of radius L, while, as usual, $S^{2}$ denotes the unit sphere in $R^{3}$ ⁠. By d_L, we denote the spherical distance on $S^{2} (L)$ ⁠. By $D (x, ρ) \subset S^{2} (L)$ ⁠, we denote the open spherical disk of radius ρ centered at x.
$\bar{G}$ is the closure of the set G.
We use the abbreviations a.s. for “almost surely,” w.o.p. for “with overwhelming probability,” which means that the property in question holds outside an event of probability O(L^−C) with every C > 0, and w.h.p. for “with high probability,” which means the property in question holds outside an event of probability O(L^−c) with some c > 0.
C and c (with or without indices) are positive constants that might only depend on the parameters in the definition of the regular Gaussian ensemble f_L (⁠ $C^{3 + ν}$ -smoothness and the power decay of correlations). One can think that the constant C is large (in particular, C ⩾ 1), while the constant c is small (in particular, c ⩽ 1). The values of these constants are irrelevant for our purposes and may vary from line to line.
A ≲ B means A ⩽ CB, A ≳ B means A ⩾ cB, and A ≃ B means that A ≲ B and A ≳ B simultaneously.
The sign ≪ means “sufficiently smaller than” and ≫ means “sufficiently larger than.” For instance, the assumption “given A and B such that A ≪ B” means that there exists c ∈ (0, 1) such that the corresponding conclusion holds for every positive A and B satisfying A ⩽ cB.

V. PRELIMINARIES

For convenience, we collect here standard facts that we will be using throughout this work.

A. Local coordinates

It will be convenient to associate with each point $p \in S^{2} (L)$ its own coordinate chart. For $p \in S^{2} (L)$ ⁠, let Π_p be a plane in $R^{3}$ passing through the origin and orthogonal to p. The Euclidean structure on Π_p is inherited from $R^{3}$ ⁠. By $S_{p}^{2} (L)$ ⁠, we denote the hemisphere of $S^{2} (L)$ centered at p. By

Ψ_{p} : \{X \in Π_{p} : | X | ⩽ L\} \to S_{p}^{2} (L),

we denote the map inverse to the orthogonal projection. Note that

| X - Y | ⩽ d_{L} (Ψ_{p} (X), Ψ_{p} (Y)) ⩽ 2 | X - Y |

whenever $| X |, | Y | ⩽ \frac{1}{2} L$ ⁠.

Let $f : S^{2} (L) \to R$ be a smooth function. We put F_p = f◦Ψ_p and identify d^kf(p) with d^kF_p(0), i.e., with a k-linear form on Π_p. Then, the gradient ∇f(p) = ∇F_p(0) is a vector in Π_p such that df(p) (v) = ⟨∇f(p), v⟩, v ∈ Π_p, and the Hessian H_f(p) = ∇²f(p) = ∇²F_p(0) is a self-adjoint operator on Π_p such that d²f(p) (u, v) = ⟨H_f(p)u, v⟩, u, v ∈ Π_p.

The notation $f \in C^{k} (S^{2} (L))$ means that, for every $p \in S^{2} (L)$ ⁠, $f ◦ Ψ_{p} \in C^{k} ({X \in Π_{p} : | X | ⩽ \frac{1}{2} L})$ ⁠, and

‖ f ‖_{C^{k}} \overset{def}{=} \sum_{j = 0}^{k} \max_{S^{2} (L)} ‖ d^{j} f ‖ .

Obviously, $‖ f ‖_{C^{k}} ⩽ \max_{p \in S^{2} (L)} ‖ f ◦ Ψ_{p} ‖_{C^{k} (\{| X | ⩽ \frac{1}{2} L)\}}$ ⁠. In the other direction, it is not difficult to see that if $f \in C^{k} (S^{2} (L))$ ⁠, then $‖ f ◦ Ψ_{p} ‖_{C^{k} (\{| X | ⩽ \frac{1}{2} L\})} ⩽ C_{k} ‖ f ‖_{C^{k}}$ ⁠.

B. Statistical properties of the gradient and the Hessian

Fix $p \in S^{2} (L)$ and the orthogonal coordinate system (X₁, X₂) on the plane Π_p, and set $\partial_{i_{1} \dots i_{k}}^{k} f (p) = \partial_{X_{i_{1}} \dots X_{i_{k}}}^{k} F_{p} (0)$ ⁠, where, as above, F_p = f◦Ψ_p.

1. Independence

To simplify the notation, next, we will deal with the case L = 1. The general case can be easily obtained by scaling.

Lemma 1.

Let f be a C^2+ν-smooth random Gaussian function on the sphere $S^{2}$ whose distribution is invariant with respect to the isometries of the sphere. Then, the following Gaussian random variables are independent:

f(p) and ∇f(p), as well as ∇f(p) and ∇²f(p),
∂₁f(p) and ∂₂f(p),
f(p) and $\partial_{1, 2}^{2} f (p)$ ⁠, and
$\partial_{1, 1}^{2} f (p)$ and $\partial_{1, 2}^{2} f (p)$ , as well as $\partial_{2, 2}^{2} f (p)$ and $\partial_{1, 2}^{2} f (p)$ .

Proof.

Without loss of generality (WLOG), we assume that p is the North Pole of the sphere

S^{2}

and suppress the dependence on p, letting Ψ = Ψ_p. Then,

Ψ (X_{1}, X_{2}) = (X_{1}, X_{2}, \sqrt{1 - (X_{1}^{2} + X_{2}^{2})}),

and F = f◦Ψ is a Gaussian function on the unit disk {|X₁|² + |X₂|² < 1} with the covariance

\begin{array}{l} K (X, Y) & = E [F (X) F (Y)] \\ = k (d (Ψ (X), Ψ (Y))) (d is the spherical distance) \\ = k (\arccos ⟨ Ψ (X), Ψ (Y) ⟩) \\ = k (\arccos (\sum_{j = 1}^{2} X_{j} Y_{j} + {(1 - \sum_{j = 1}^{2} X_{j}^{2})}^{\frac{1}{2}} {(1 - \sum_{j = 1}^{2} Y_{j}^{2})}^{\frac{1}{2}})) . \end{array}

Note that

K (- X_{1}, X_{2}, - Y_{1}, Y_{2}) = K (X_{1}, - X_{2}, Y_{1}, - Y_{2}) = K (X_{1}, X_{2}, Y_{1}, Y_{2}) .

(5.1)

To prove properties (i)–(iv), we need to check the corresponding statements for F(0),

\partial_{X_{j}} F (0)

⁠, and

\partial_{X_{i} X_{j}}^{2} F (0)

⁠. The covariances of these random variables can be computed using the relations

E [\frac{\partial^{ℓ} F (X)}{\partial^{ℓ_{1}} X_{1} \partial^{ℓ_{2}} X_{2}} \frac{\partial^{m} F (Y)}{\partial^{m_{1}} Y_{1} \partial^{m_{2}} Y_{2}}] = \frac{\partial^{ℓ + m} K (X, Y)}{\partial^{ℓ_{1}} X_{1} \partial^{ℓ_{2}} X_{2} \partial^{m_{1}} Y_{1} \partial^{m_{2}} Y_{2}} .

The rest follows by differentiation of relations (). For instance,

\frac{\partial^{2} K}{\partial Y_{1} \partial Y_{2}} (X_{1}, X_{2}, Y_{1}, Y_{2}) = - \frac{\partial^{2} K}{\partial Y_{1} \partial Y_{2}} (- X_{1}, X_{2}, - Y_{1}, Y_{2}),

whence

E [F (0) \partial_{Y_{1} Y_{2}}^{2} F (0)] = \frac{\partial^{2} K}{\partial Y_{1} \partial Y_{2}} (0, 0) = 0 .

Similarly,

\frac{\partial^{4} K}{\partial^{2} X_{1} \partial Y_{1} \partial Y_{2}} (X_{1}, X_{2}, Y_{1}, Y_{2}) = - \frac{\partial^{4} K}{\partial^{2} X_{1} \partial Y_{1} \partial Y_{2}} (- X_{1}, X_{2}, - Y_{1}, Y_{2}),

whence

E [\partial_{X_{1}}^{2} F (0) \partial_{Y_{1} Y_{2}}^{2} F (0)] = \frac{\partial^{4} K}{\partial_{X_{1}}^{2} \partial Y_{1} \partial Y_{2}} (0, 0) = 0

and so on.□

2. Non-degeneracy

Lemma 2.

Let (f_L) be a regular Gaussian ensemble, and let

p \in S^{2} (L)

. Then,

\underset{L \to \infty}{lim inf} E [{(\partial_{j} f_{L} (p))}^{2}] > 0, j \in {1, 2},

and

\underset{L \to \infty}{lim inf} E [{(\partial_{i, j}^{2} f_{L} (p))}^{2}] > 0, i, j \in {1, 2} .

Proof.

Again, we assume that p is the North Pole of the sphere $S^{2} (L)$ ⁠. Put $Ψ_{L} (X_{1}, X_{2}) = (X_{1}, X_{2}, \sqrt{L^{2} - (X_{1}^{2} + X_{2}^{2})})$ ⁠, and consider the Gaussian functions F_L = f_L◦Ψ_L defined in the disks $\frac{1}{2} L D$ ⁠. The corresponding covariances $E [F_{L} (X) F_{L} (Y)] = K_{L} (X, Y)$ are C^3+ν,3+ν-smooth on $\frac{1}{2} L D \times \frac{1}{2} L D$ with some ν > 0. Their partial derivatives up to the third order are bounded locally uniformly in L ⩾ L₀. Hence, by a version of the Arzelá–Ascoli theorem, any sequence $K_{L_{j}}$ contains a locally uniformly C^2+ν,2+ν-convergent subsequence.

The limiting function K is a C^2+ν,2+ν-smooth Hermitian-positive function on

R^{2} \times R^{2}

⁠, which depends only on the Euclidean distance |X–Y|. Hence, by Bochner’s theorem, it is a Fourier integral of a finite positive rotation-invariant measure ρ,

K (X, Y) = \int_{R^{2}} e^{2 π i ⟨ λ, X - Y ⟩} d ρ (λ),

where

\int_{R^{2}} | λ |^{4} d ρ (λ) < \infty .

Furthermore, since

|K_{L} (X, Y)| ≲ {(1 + d_{L} (Ψ_{L} (X), Ψ_{L} (Y)))}^{- γ},

uniformly in L ⩾ L₀, the limiting function K satisfies

|K (X, Y)| ≲ {(1 + | X - Y |)}^{- γ} .

Therefore, the measure ρ cannot degenerate to the point measure at the origin.

The rest is straightforward. Suppose, for instance, that for some sequence L_j → ∞,

\lim_{j \to \infty} E [{(\partial_{1} f_{L_{j}} (p))}^{2}] = 0 .

Then,

\lim_{j \to \infty} \frac{\partial^{2} K_{L_{j}}}{\partial X_{1} \partial Y_{1}} (0, 0) = 0 .

Passing to a subsequence, we conclude that

{(2 π i)}^{2} \int_{R^{2}} λ_{1}^{2} d ρ (λ) = - \frac{\partial^{2}}{\partial X_{1} \partial Y_{1}} \int_{R^{2}} e^{2 π i 〈 λ, X - Y 〉} d ρ (λ) {|)}_{X = Y = 0} = 0 .

Since the measure ρ is positive and rotation-invariant, this is possible only when ρ is a point mass at the origin. This contradiction concludes the proof.□

3. Power decay of correlations

The power decay of the correlations between f_L(p) and f_L(q) when d_L(p, q) is large and the a priori C^3,3-smoothness of the covariance yield the power decay of correlations between the Gaussian vectors

v (p) = (f_{L} (p), \nabla f_{L} (p)) = (F_{p} (0), \nabla F_{p} (0)) = (F_{p} (0), \partial_{X_{1}} F_{p} (0), \partial_{X_{2}} F_{p} (0))

and

v (q) = (f_{L} (q), \nabla f_{L} (q)) = (F_{q} (0), \nabla F_{q} (0)) = (F_{q} (0), \partial_{Y_{1}} F_{q} (0), \partial_{Y_{2}} F_{q} (0))

[we keep fixed the coordinate systems (X₁, X₂) and (Y₁, Y₂) in the planes Π_p and Π_q].

Lemma 3.

Let (f_L) be a regular Gaussian ensemble. Then, for any

p, q \in S^{2} (L)

,

\max_{1 ⩽ i, j ⩽ 3} |E [v_{i} (p) v_{j} (q)]| ≲ {(1 + d_{L} (p, q))}^{- γ / 4} .

Proof.

Put $K_{p, q} (X, Y) = E [F_{p} (X) F_{q} (Y)]$ ⁠, where F_p = f_L◦Ψ_p and $(X, Y) \in \bar{Q} \times \bar{Q}$ ⁠, where $\bar{Q} = [- 1, 1] \times [- 1, 1]$ ⁠. We have $E [F_{p} (0) \partial_{Y_{i}} F_{q} (0)] = \partial_{Y_{i}} K_{p, q} (0, 0)$ ⁠, and $E [\partial_{X_{i}} F_{p} (0) \partial_{Y_{j}} F_{q} (0)] = \partial_{X_{i} Y_{j}}^{2} K_{p, q} (0, 0)$ ⁠.

There is nothing to prove if d_L(p, q) ⩽ 1, so we assume that d_L(p, q) ⩾ 1. Then, $‖ K_{p, q} ‖_{C (\bar{Q} \times \bar{Q})} ≲ d^{- γ}$ and $‖ K_{p, q} ‖_{C^{2, 2} (\bar{Q} \times \bar{Q})} ≲ 1$ ⁠. Now, we use the classical Landau–Hadamard inequality in the following form: if $h : [0, 1] \to R$ is a C²-smooth function and M_j = max_[0,1]|h^(j)|, 0 ⩽ j ⩽ 2, then $M_{1} ≲ \max (M_{0}, \sqrt{M_{0} M_{2}})$ ⁠. Applying this to the functions X_i↦K_p,q(X, Y) and Y_j↦K_p,q(X, Y), we get $‖ \partial_{X_{i}} K_{p, q} ‖_{C (\bar{Q} \times \bar{Q})} ≲ d^{- γ / 2}$ and $‖ \partial_{Y_{j}} K_{p, q} ‖_{C (\bar{Q} \times \bar{Q})} ≲ d^{- γ / 2}$ ⁠, i, j = 1, 2. Applying the Landau–Hadamard inequality again, this time to the functions $Y_{j} \mapsto \partial_{X_{i}} K_{p, q} (X, Y)$ ⁠, we get $‖ \partial_{X_{i} Y_{j}}^{2} K_{p, q} ‖_{C (\bar{Q} \times \bar{Q})} ≲ d^{- γ / 4}$ ⁠, i, j = 1, 2.

In particular, these estimates hold at X = Y = 0, which gives us what we needed.□

To simplify our notation, in the following, we assume that the parameter γ > 0 is chosen so that the correlations between the Gaussian vectors (f_L(p), ∇f_L(p)) and (f_L(q), ∇f_L(q)) decay as ${(1 + d_{L} (p, q))}^{- γ}$ ⁠.

C. A priori smoothness of f_L

Let (f_L) be a regular Gaussian ensemble. Quite often, we will be using the following a priori bound:

w.o.p, $‖ f_{L} ‖_{C^{3} (S^{2} (L))} < \log L$ ⁠.

This bound immediately follows from the classical estimate

P \{‖ f_{L} ‖_{C^{3} (S^{2} (L))} > t\} ⩽ C L^{2} e^{- c t^{2}} .

For a self-contained proof, see Secs. A9–A11 of Ref. 6.

VI. SMOOTH FUNCTIONS WITH CONTROLLED TOPOLOGY OF THE ZERO SET

Here, we introduce the (non-random) class C³(A, Δ, α, β) of smooth functions f on $S^{2} (L)$ such that the number of connected components of the zero set of a small perturbation $\tilde{f}$ of f can be recovered from the values of $\tilde{f}$ at the critical points of f with small critical values (provided that the values of $\tilde{f}$ at these points are not too small). Later, we will show that w.h.p., our random function f_L belongs to this class.

A. The sets Cr(α), Cr(α, β), and Cr(α, β, Δ) and the class C³(A, Δ, α, β)

Given α, β ⩽ 1 and Δ ⩾ 1, we let

Cr (α) = {p \in S^{2} (L) : | f (p) | ⩽ α, \nabla f (p) = 0},

Cr (α, β) = {p \in S^{2} (L) : | f (p) | ⩽ α, | \nabla f (p) | ⩽ β},

and

Cr (α, β, Δ) = {p \in S^{2} (L) : | f (p) | ⩽ α, | \nabla f (p) | ⩽ β, ‖ {(\nabla^{2} f (p))}^{- 1} ‖_{op} ⩽ Δ},

where ‖⋅‖_6op stands for the operator norm.

By C³(A), we denote the class of C³-smooth functions f on $S^{2} (L)$ with $‖ f ‖_{C^{3}} ⩽ A$ ⁠. Given the parameters

α ≪ β ≪ 1 ≪ A ≪ Δ,

by C³(A, Δ, α, β), we denote the class of functions f ∈ C³(A) for which Cr(α, β) = Cr(α, β, Δ), i.e., the Hessian of f does not degenerate [ $‖ {(\nabla^{2} f)}^{- 1} ‖_{op} ⩽ Δ$ ] on the almost singular set Cr(α, β) where f and ∇f are simultaneously small.

Given f ∈ C³(A, Δ, α, β), α′ ≪ α, and g ∈ C³(A), we set

f_{t} = f + t g, 0 ⩽ t ⩽ α^{'} .

Next, we develop a little caricature of the quantitative Morse theory, which shows that the collection of signs of f_t at Cr(α) defines the topology of the zero set Z(f_t), provided that min_Cr(α)|f_t| is not too small, and gives “an explicit formula” that recovers the number of connected components of Z(f_t) from this collection of signs and the structure of Z(f).

B. Near any almost singular point there is a unique critical point of f

Lemma 4.

Suppose that f ∈ C³(A) and p ∈ Cr(α, β, Δ) with

1 ≪ A ≪ Δ, A Δ^{2} β ≪ 1 .

Then,

the spherical disk D(p, 2Δβ) contains a unique critical point z of f,
there are no other critical points of f in the disk $D (p, c {(A Δ)}^{- 1})$ , and
|f(z)| ⩽ 2α, provided that AΔ²β² ≪ α.

Proof.

We will work on the plane Π_p, and let F = f◦Ψ_p and

H_{F} = \nabla^{2} F

⁠. To find the critical point z, we use a simplified Newton method,

X_{n + 1} = X_{n} - H_{F} {(0)}^{- 1} \nabla F (X_{n}), X_{0} = 0 .

Put

Φ (X) = X - H_{F} {(0)}^{- 1} \nabla F (X)

⁠. First, we check that in the disk

D (0, c {(A Δ)}^{- 1})

⁠, the map Φ is a

\frac{1}{3}

-contraction.

Indeed,

\begin{array}{l} Φ (X) - Φ (Y) = (X - Y) & - H_{F} {(0)}^{- 1} (\nabla F (X) - \nabla F (Y)) \\ = (X - Y) - H_{F} {(0)}^{- 1} [H_{F} (X) (X - Y) + O (A | X - Y |^{2})], \end{array}

whence

| Φ (X) - Φ (Y) | ⩽ ‖ I - H_{F} {(0)}^{- 1} H_{F} (X) ‖_{op} | X - Y | + Δ O (A | X - Y |^{2}) .

Furthermore,

‖ I - H_{F} {(0)}^{- 1} H_{F} (X) ‖_{op} ⩽ ‖ H_{F} {(0)}^{- 1} ‖_{op} ‖ H_{F} (0) - H_{F} (X) ‖_{op} ⩽ Δ \cdot O (A | X |),

and finally,

| Φ (X) - Φ (Y) | ⩽ A Δ \cdot (O (| X |) + O (| X - Y |)) | X - Y | ⩽ \frac{1}{3} | X - Y |,

provided that |X|, |Y| ⩽ c(AΔ)⁻¹ with sufficiently small positive c.

Next, we check that the map Φ preserves the disk D(0, c(AΔ)⁻¹). We have

\begin{array}{l} Φ (X) & = X - H_{F} {(0)}^{- 1} \nabla F (X) \\ = - H_{F} {(0)}^{- 1} (\nabla F (X) - H_{F} (0) X) = - H_{F} {(0)}^{- 1} (\nabla F (0) + O (A | X |^{2})), \end{array}

and then,

| Φ (X) | ⩽ Δ (β + A {(\frac{c}{A Δ})}^{2} O (1)) = β Δ + \frac{c^{2}}{A Δ} O (1) < \frac{c}{A Δ},

provided that βAΔ² ≪ 1 and that the positive constant c is sufficiently small.

Thus, Φ has a unique fixed point Z in the disk D(0, c(AΔ)⁻¹), and

\begin{array}{l} | Z | = | 0 - Z | & ⩽ \sum_{n ⩾ 0} | X_{n} - X_{n + 1} | ⩽ \sum_{n ⩾ 0} 3^{- n} | X_{0} - X_{1} | \\ = \frac{3}{2} | 0 - Φ (0) | = \frac{3}{2} | Φ (0) | = \frac{3}{2} | H_{F} {(0)}^{- 1} \nabla F (0) | ⩽ \frac{3}{2} Δ β . \end{array}

At last,

| F (Z) | ⩽ | F (0) | + | \nabla F (0) | | Z | + O (A | Z |^{2}) ⩽ α + β \cdot 2 β Δ + O (A β^{2} Δ^{2}) ⩽ 2 α,

provided that Aβ²Δ² ≪ α.□

C. Near any point p ∈ Cr(α) there is a unique critical point of f_t

Lemma 5.

Let f, g ∈ C³(A), where f_t = f + tg with 0 ⩽ t ⩽ α′. Suppose that p ∈ Cr(α) with

‖ H_{f} {(p)}^{- 1} ‖_{op} ⩽ Δ

and that

1 ≪ A ≪ Δ, A α^{'} ≪ α, A Δ^{2} α ≪ 1 .

Then, there exists a unique critical point p_t of f_t such that d_L(p, p_t) ≪ Δα and |f_t(p) − f_t(p_t)| ≪ A(Δα)². Moreover, there are no other critical points of f_t at distance ⩽c(AΔ)⁻¹ from p.

Proof of Lemma 5.

First, we note that f_t ∈ C³(2A) and that |f_t(p)| ⩽ |f(p)| + Aα′ and |∇f_t(p)|≲ Aα′ ≪ α. Furthermore,

‖ H_{f_{t}} {(p)}^{- 1} ‖_{op} ⩽ 2 Δ

⁠. Indeed, we have

\begin{array}{l} ‖ H_{f_{t}} {(p)}^{- 1} ‖_{op} & = ‖ H_{f} {(p)}^{- 1} {(I + (H_{f_{t}} (p) - H_{f} (p)) H_{f} {(p)}^{- 1})}^{- 1} ‖_{op} \\ ⩽ Δ ‖ {(I + (H_{f_{t}} (p) - H_{f} (p)) H_{f} {(p)}^{- 1})}^{- 1} ‖_{op} . \end{array}

Noting that

‖ H_{f_{t}} (p) - H_{f} (p) ‖_{op} ≲ A α^{'}

⁠, we see that

‖ (H_{f_{t}} (p) - H_{f} (p)) H_{f} {(p)}^{- 1} ‖_{op} ≲ Δ \cdot A α^{'} ≪ Δ α ≪ 1 .

Therefore,

‖ {(I + (H_{f_{t}} (p) - H_{f} (p)) H_{f} {(p)}^{- 1})}^{- 1} ‖_{op} < 2,

and finally,

‖ H_{f_{t}} {(p)}^{- 1} ‖_{op} ⩽ 2 Δ

⁠.

Then, by Lemma 4 (applied to the function f_t with β = Aα′), there exists a unique critical point p_t of f_t with

d_{L} (p, p_{t}) ⩽ 2 (A α^{'}) \cdot 2 Δ ≪ Δ α .

Besides, for d_L(p, x) ≪ Δα, we have d_L(p_t, x) ≪ Δα and then |∇f_t(x)| = |∇f_t(x) − ∇f_t(p_t)| ≪ A ⋅ Δα, whence

| f_{t} (p) - f_{t} (p_{t}) | ≪ A {(Δ α)}^{2} .

At last, by part B of Lemma 4, there are no other critical points of f_t at distance ⩽c(AΔ)⁻¹ from p.□

D. Local matters

Given f ∈ C³(A), p ∈ Cr(α), and $‖ H_{f} {(p)}^{- 1} ‖_{op} ⩽ Δ$ ⁠, we look at the behavior of f_t in the δ-neighborhood of p. As above, f_t = f + tg, with g ∈ C³(A) and 0 ⩽ t ⩽ α′. Throughout this section, we assume that the parameters α′, α, δ, A, and Δ satisfy the set of conditions

α ≪ 1 ≪ A ≪ Δ, A α^{'} ≪ α ≪ A^{- 2} Δ^{- 3}

(6.1)

and that

δ = c {(A Δ)}^{- 1},

(6.2)

with sufficiently small constant c. Note that these conditions are more restrictive than the ones used in Lemma 5, so we will be using freely that lemma.

1. Local extrema

First, we consider the case when the Hessian $H_{f} (p)$ is positive or negative definite, that is, its eigenvalues have the same sign. With a little abuse of terminology, we say that the function f_t is convex (concave) in D(p, δ) if the function f_t◦Ψ_p is convex (correspondingly, concave) in $Ψ_{p}^{- 1} D (p, δ) \subset Π_{p}$ ⁠.

Lemma 6.

Suppose that the eigenvalues of the Hessian $H_{f} (p)$ have the same sign and that conditions (6.1) and (6.2) hold. Then,

the function f_t is either concave or convex function in D(p, δ), and
the function f_t does not vanish on ∂D(p, δ), and moreover, the sign of $f_{t} {|)}_{\partial D (p, δ)}$ coincides with the sign of the eigenvalues of $H_{f} (p)$ .

Proof of Lemma 6.

Put F = f◦Ψ_p and F_t = f_t◦Ψ_p, and suppose that

H_{F} (0) = H_{f} (p)

is positive definite (otherwise, replace f by −f), that is,

〈 H_{F} (0) x, x 〉 ⩾ Δ^{- 1} | x |^{2}

⁠. Then, for any

X \in Ψ_{p}^{- 1} D (p, δ)

⁠, we have

\begin{array}{l} 〈 H_{F} (X) x, x 〉 ⩾ 〈 H_{F} (0) x, x 〉 - ‖ H_{F} (X) & - H_{F} (0) ‖_{op} | x |^{2} ⩾ (Δ^{- 1} - C A | X |) | x |^{2} ⩾ {(2 Δ)}^{- 1} | x |^{2} \end{array}

since

‖ H_{F} (X) - H_{F} (0) ‖_{op} ≲ A | X |

and |X| ⩽ δ = c(AΔ)⁻¹ with sufficiently small c. Noting that

‖ H_{F_{t}} (X) - H_{F} (X) ‖_{op} ≲ A α^{'} ≪ Δ^{- 1}

⁠, we get

〈 H_{F_{t}} (X) x, x 〉 ⩾ {(4 Δ)}^{- 1} | x |^{2}

⁠, which proves (i).

To prove (ii), we take

X \in Ψ_{p}^{- 1} \partial D (p, δ)

⁠. Then,

\begin{array}{l} F_{t} (X) = F_{t} (0) + 〈 \nabla F_{t} (0), X 〉 & + \frac{1}{2} 〈 H_{F_{t}} (0) X, X 〉 + O (A δ^{3}) \\ ⩾ \frac{1}{2} 〈 H_{F_{t}} (0) X, X 〉 - | F_{t} (0) | - | 〈 \nabla F_{t} (0), X 〉 | - O (A δ^{3}) . \end{array}

Furthermore, using that |F_t(0)| = |f_t(p)| ⩽ α + O(Aα′) ≲ α and that |⟨∇F_t(0), X⟩| ⩽ |∇F_t(0)|⋅|X| = O(α′A) ⋅ δ ≪ α, we conclude that

\begin{array}{l} F_{t} (X) & \overset{| X | ⩾ δ / 2}{⩾} \frac{1}{2} {(4 Δ)}^{- 1} {(δ / 2)}^{2} - O (α + A δ^{3}) = {(32 Δ)}^{- 1} δ^{2} - O (A δ^{3}) \\ = ({(32 Δ)}^{- 1} - O (A δ)) δ^{2} \overset{δ = c {(A Δ)}^{- 1}}{⩾} (\frac{1}{32} - C \cdot c) Δ^{- 1} δ^{2} ⩾ \frac{1}{64} Δ^{- 1} δ^{2}, \end{array}

provided that the constant c in () (the definition of δ) was chosen so small that

C \cdot c ⩽ \frac{1}{64}

⁠.□

Summary: Let p be a local extremum of f. Suppose that conditions (6.1) and (6.2) hold.

Then, Z(f_t) ∩ ∂D(p, δ) = ∅.
Z(f_t) ∩ D(p, δ) is either empty, homeomorphic to $S^{1}$ ⁠, or a singleton.
Suppose that |f_t(p)| ≳ A(Δα)². Then, by Lemma 5, f_t(p_t) has the same sign as f_t(p). Therefore, Z(f_t) ∩ D(p, δ) = ∅ whenever f_t(p) and the eigenvalues of $H_{f} (p)$ have the same sign, and Z(f_t) ∩ D(p, δ) is homeomorphic to $S^{1}$ whenever f_t(p) and the eigenvalues of $H_{f} (p)$ have opposite signs.

2. Saddle points

Now, we turn to the case when p ∈ Cr(α) is a saddle point of f, that is, the eigenvalues of $H_{f} (p)$ have opposite signs. We will work on the plane Π_p and set F = f◦Ψ_p, G = g◦Ψ_p, and F_t = f_t◦Ψ_p = F + tG. By $H (X) = ⟨ H_{F} (0) X, X ⟩$ ⁠, we denote the quadratic form generated by the Hessian $H_{F} (0)$ ⁠. WLOG, we assume that

H (X) = a X_{1}^{2} - b X_{2}^{2}, Δ^{- 1} ⩽ a ⩽ b ⩽ A .

We take δ = c(AΔ)⁻¹ with a sufficiently small positive constant c, set

J (δ) \overset{def}{=} \{| H | ⩽ a δ^{2}\} ⋂ \{| X_{1} | ⩽ 3 δ\},

and call this set a joint. By

\partial^{*} J (δ) \overset{def}{=} \{| H | = a δ^{2}\} ⋂ \{| X_{1} | ⩽ 3 δ\},

we denote the curvilinear part of the full boundary ∂J(δ) of the joint J(δ) (Fig. 2).

FIG. 2.

View large Download slide

Joint J(δ) with four terminals.

Lemma 7.

Suppose that the eigenvalues of $H_{f} (p)$ have the opposite signs and that conditions (6.1) and (6.2) hold. Then, the function F_t does not vanish on ∂*J(δ). Moreover, the signs of F_t and H coincide on ∂*J(δ).

Proof of Lemma 7.

Everywhere in J(δ), we have

F_{t} = F (0) + \frac{1}{2} H + O (A δ^{3}) + α^{'} G = \frac{1}{2} H + O (α + (δ^{3} + α^{'}) A) \overset{α^{'} ≪ δ^{3}}{=} \frac{1}{2} H + O (A δ^{3}) .

Furthermore, on ∂*J(δ), we have |H| = aδ² ⩾ c²A⁻²Δ⁻³, while Aδ³ = c³A⁻²Δ⁻³. This proves the lemma.□

The set $\{| H | ⩽ a δ^{2}\} ⋂ \{2 δ ⩽ | X_{1} | ⩽ 3 δ\}$ consists of four disjoint curvilinear quadrangles. We call them terminals and denote them by T_i, 1 ⩽ i ⩽ 4.

Lemma 8.

Under the same assumptions as in Lemma 7, each of the sets Z(F_t) ∩ T_i consists of one curve, which joins the vertical segments on the boundary of T_i.

Proof of Lemma 8.

Everywhere in J(δ), we have

{(F_{t})}_{X_{2}} = \frac{1}{2} H_{X_{2}} + (F_{X_{2}} - \frac{1}{2} H_{X_{2}}) + t G_{X_{2}} .

Hence,

| {(F_{t})}_{X_{2}} | ⩾ b | X_{2} | - C A | X |^{2} - O (A α^{'}) .

In each of the terminals T_i,

b | X_{2} | ⩾ b \cdot \sqrt{3 \frac{a}{b}} \cdot δ > \sqrt{a b} \cdot δ ⩾ \frac{δ}{Δ}

(in the last estimate, we use that b ⩾ a ⩾ Δ⁻¹), and

| X |^{2} ⩽ {(3 δ)}^{2} + {(\sqrt{10 \frac{a}{b}} \cdot δ)}^{2} ⩽ 19 δ^{2},

whence

C A | X |^{2} ⩽ 19 C A δ^{2} = 19 C A \frac{c}{A Δ} \cdot δ ⩽ \frac{δ}{2 Δ},

provided that the constant c in the definition of δ is sufficiently small. Furthermore, Aα′ is also much smaller than Δ⁻¹δ (since Aα′ ≪ A⁻¹Δ⁻²). Thus,

| {(F_{t})}_{X_{2}} | > 0

everywhere in T_i. It remains to recall that, by Lemma 7, the function F_t has at least one change in sign on each vertical section of T_i. Therefore, by the implicit function theorem, Z(F_t) ∩ T_i is a graph of a smooth function.□

Under the same assumptions as in Lemmas 7 and 8, by Lemma 5, the joint J(δ) contains only one critical point $X^{t} = (X_{1}^{t}, X_{2}^{t})$ of F_t and |X^t| ≪ Δα. Consider the sets

I_{1} = \{X = (X_{1}, X_{2}^{t}) : X_{1} \in R\} \cap J (δ), I_{2} = \{X = (X_{1}^{t}, X_{2}) : X_{2} \in R\} \cap J (δ) .

Since

\begin{array}{l} | X^{t} | ≪ Δ α ≪ {(A Δ)}^{- 2} = c^{- 1} {(A Δ)}^{- 1} δ ≪ {(A Δ)}^{- 1 / 2} δ ⩽ \sqrt{a / b} \cdot δ (Δ^{- 1} ⩽ a ⩽ b ⩽ A), \end{array}

it is easy to see that both sets are the segments.

Lemma 9.

Under the same assumptions as in Lemmas 7 and 8, the only extremum of the restriction of the function F_t to the segment I₁ is a local minimum at $X_{1} = X_{1}^{t}$ , and the only extremum of the restriction of the function F_t to the segment I₂ is a local maximum at $X_{2} = X_{2}^{t}$ .

Proof of Lemma 9.

Consider the function

X_{1} \mapsto F_{t} (X_{1}, X_{2}^{t})

⁠. It has a critical point at

X_{1} = X_{1}^{t}

⁠, the second derivative at this point is not less than

2 a - C A α^{'} - C A | X^{t} | ⩾ \frac{2}{Δ} - C A (α^{'} + Δ α) ⩾ \frac{1}{Δ}

(since AΔ²α ≪ 1, and Aα′ ≪ α), and the C³-norm of F_t is bounded by CA. Therefore, for

X_{1} > X_{1}^{t}

⁠, we have

{(F_{t})}_{X_{1}} (X_{1}, X_{2}^{t}) ⩾ 2 a (X_{1} - X_{1}^{t}) - C A {(X_{1} - X_{1}^{t})}^{2} ⩾ Δ^{- 1} (X_{1} - X_{1}^{t}) - C A | X_{1} - X_{1}^{t} |^{2} .

Note that the RHS of the last expression is positive since

X_{1} - X_{1}^{t} < 2 δ = 2 c {(A Δ)}^{- 1}

with sufficiently small constant c. Similarly,

{(F_{t})}_{X_{1}} (X_{1}, X_{2}^{t}) < 0

for

X_{1} < X_{1}^{t}

⁠.

The proof of the second statement is almost identical and we skip it.□

Lemma 10.

Suppose that F_t(X^t) ≠ 0 [i.e., zero is not a critical value of the restriction of the function F_t to the joint J(δ)]. Then, under the same assumptions as in Lemmas 7–9, the set Z(F_t) ∩ J(δ) consists of two connected components, which enter and exit the joint J(δ) through the terminals T_i.

Furthermore, the set {F_t ≠ 0} ∩ J(δ) consists of three connected components. One of them contains F_t(X^t), while on the other two components, F_t has the sign opposite to the sign of F_t(X^t).

Proof of Lemma 10.

Since zero is not a critical value of the restriction $F_{t} {|)}_{J (δ)}$ ⁠, the set Z(F_t) ∩ J(δ) consists of a finitely many disjoint smooth curves. By Lemma 8, this set has at least two connected components, the ones that enter and exit the joint J(δ) through the terminals. If there exists a third component, then, again by Lemma 8, it cannot intersect the terminals, while by Lemma 7, it also cannot intersect the rest of the boundary ∂*J(δ). Hence, it stays inside the joint. Therefore, it is a closed curve, which bounds a domain G with $\bar{G} \subset J (δ)$ ⁠. Since F_t vanishes on ∂G, G must contain the (unique) critical point X^t of F_t and ∂G separates X^t from ∂J(δ). On the other hand, Lemma 9 together with Lemma 7 yields that on one of the segments I_i, i = 1, 2, the function F_t does not change its sign. The resulting contradiction proves the first part of the lemma.

To prove the second part, first, we notice that, since the sets {F_t > 0} ∩ J(δ) and {F_t < 0} ∩ J(δ) cannot be simultaneously connected, the set {F_t ≠ 0} ∩ J(δ) has at least three connected components. One of them, we call it Ω₀, contains the critical point X^t, and therefore, by Lemma 9, it contains one of the segments I_i. Since F_t does not vanish on ∂*J(δ) (Lemma 7), the boundary of Ω₀ contains two opposite sides of ∂*J(δ), the ones on which the end points of the segment I_i lie. For the same reason, there are two more connected components of the set {F_t ≠ 0} ∩ J(δ), and each of these two components contains on its boundary one of two remaining opposite sides of the set ∂*J(δ). At last, arguing as in the proof of the first part (also using again Lemmas 8 and 9), we see that the fourth connected component of the set {F_t ≠ 0} ∩ J(δ) cannot exist.□

Summary: Let p be a saddle point of f. Suppose that conditions (6.1) and (6.2) hold, and let J(p, δ) = Ψ_pJ(δ) be the corresponding joint. Suppose that 0 is not a critical value of f_t.

Then, the set Z(f_t) ∩ J(p, δ) consists of two connected components. Each of them enters and exits the joint through its own terminals Ψ_pT_i.
We say that the joint J(p, δ) has positive type if the set J(p, δ) ∩{f_t > 0} is connected [therefore, the set J(p, δ) ∩{f_t < 0} is disconnected and consists of two connected components]. Otherwise, we say that the joint J(p, δ) has negative type. Suppose that |f_t(p)|≳ A(Δα)². Then, the type of the joint J(p, δ) coincides with the sign of f_t(p).

E. Global matters: The gradient flow

Fix the functions f ∈ C³(A, Δ, α, β) and g ∈ C³(A). Let f_t = f + tg and $\tilde{f} = f_{α^{'}}$ ⁠, and consider the gradient flow z_t, 0 ⩽ t ⩽ α′, defined by the ordinary differential equation (ODE),

\frac{d z_{t}}{d t} = - \frac{(\partial_{t} f_{t}) (z_{t})}{| \nabla f_{t} (z_{t}) |^{2}} \nabla f_{t} (z_{t})

(6.3)

with the initial condition z₀ ∈ Z(f).

Lemma 11.

Suppose that AΔ²β² ≪ α ≪ (AΔ)⁻²β and Aα′ ≪ α. Let δ = c(AΔ)⁻¹ with a sufficiently small constant c > 0. Then, for any arc I ⊂ Z(f) \⋃_p∈Cr(α)D(p, 2δ²), the flow z_t provides a C¹-homotopy of I onto an arc $\tilde{I} \subset Z (\tilde{f}) \ ⋃_{p \in Cr (α)} D (p, δ^{2})$ . Vice versa, for any arc $\tilde{I} \subset Z (\tilde{f}) \ ⋃_{p \in Cr (α)} D (p, 2 δ^{2})$ , the inverse flow z_α′−t provides a C¹-homotopy of $\tilde{I}$ onto an arc I ⊂ Z(f) \⋃_p∈Cr(α)D(p, δ²). Moreover, these homotopies move the points by at most O(Aα′/β).

Proof of Lemma 11.

Let

X \overset{def}{=} \{| \nabla f | < β\}

and

\bar{Ω} = \bar{Ω} (ε, η) \overset{def}{=} \{(t, x) \in [- ε, α^{'} + ε] \times (S^{2} (L) \ X) : | f_{t} (x) | ⩽ η\}

(the choice of small positive parameters ɛ and η has no importance), and let Ω be the interior of

\bar{Ω}

⁠. Note that

| \partial_{t} f_{t} | = | g | ⩽ A everywhere,

and

| \nabla f_{t} | ⩾ | \nabla f | - A α^{'} ⩾ \frac{1}{2} β on S^{2} (L) \ X .

Therefore, the flow moves the points with the speed

|\frac{d z_{t}}{d t}| = |\frac{(\partial_{t} f_{t}) (z_{t})}{| \nabla f_{t} (z_{t}) |}| ⩽ \frac{2 A}{β} .

The RHS of the ODE () is a C¹-function on

\bar{Ω}

⁠. Therefore, for any initial point

z_{0} \in Z (f) \ \bar{X}

⁠, the ODE has a unique C¹-solution. The solution exists until it reaches the boundary of Ω. Note that along the trajectory z_t, we have

\frac{d}{d t} f_{t} (z_{t}) = (\frac{\partial}{\partial t} f_{t}) (z_{t}) + ⟨\nabla f_{t} (z_{t}), \frac{d z_{t}}{d t}⟩ = 0,

whence f_t(z_t) = 0 [recall that z₀ ∈ Z(f)]. Hence, if the solution z_t is not defined on [0, α′], then there exists τ ⩽ α′ such that d_L(z_t, X) → 0 as t ↑ τ. This means that the closure of the trajectory z_t, 0 ⩽ t < τ, contains a point

\bar{z}

with

| \nabla f (\bar{z}) | ⩽ β

⁠. Recalling that the point z_t moves with the speed ⩽ 2A/β, we see that

d_{L} (\bar{z}, z_{0}) ⩽ 2 A α^{'} / β

⁠. Furthermore, by the continuity of f_t, we have

f_{τ} (\bar{z}) = 0

⁠, whence

| f (\bar{z}) | ⩽ α^{'} | g (\bar{z}) | ⩽ A α^{'} ≪ α

⁠. Combining this with the gradient estimate

| \nabla f (\bar{z}) | ⩽ β

and applying Lemma 4, we conclude that there is a unique critical point p of f with

d_{L} (p, \bar{z}) ⩽ 2 β Δ

⁠, i.e., with

d_{L} (p, z_{0}) ⩽ 2 β Δ + \frac{2 A α^{'}}{β},

which is much less than δ².

It remains to check that this critical point p belongs to Cr(α), which is straightforward,

| f (p) | ≲ | f (\bar{z}) | + d_{L} (\bar{z}, p) β + O (d_{L} {(\bar{z}, p)}^{2} A) ≲ A α^{'} + Δ β^{2} + A Δ^{2} β^{2} ≪ α,

since Aα′ and AΔ²β² are both much less than α.

Since the function $\tilde{f} = f + α^{'} g$ belongs to the class C³(2A, 2Δ, 2α, 2β), the same arguments can be also applied to the inverse flow z_α′−t.□

F. The upshot

We start with functions f ∈ C³(A, Δ, α, β) and g ∈ C³(A) and consider the perturbation $\tilde{f} = f + α^{'} g$ ⁠. We assume that the parameters

α^{'} ≪ α ≪ β ≪ 1 ≪ A ≪ Δ

satisfy the following relations:

A α^{'} ≪ α, A Δ^{2} β^{2} ≪ α ≪ {(A Δ)}^{- 2} β, A^{2} Δ^{3} α ≪ 1

[which particularly yield conditions (6.1)]. We also assume that the perturbation $\tilde{f}$ is not too small on Cr(α),

\min_{Cr (α)} | \tilde{f} | ≳ A Δ^{2} α^{2} .

We set

\begin{array}{l} {Cr}_{S} (α) & = {p \in Cr (α) : p is a saddle point of f}, \\ {Cr}_{E} (α) & = {p \in Cr (α) : p is a local extremum of f} . \end{array}

We put δ = c(AΔ)⁻¹ with sufficiently small positive constant c and consider the disks D(p, δ) and p ∈ Cr_E(α) and the joints J(p, δ) and p ∈ Cr_S(α). If the constant c in the definition of δ was chosen sufficiently small, then all these disks and joints are mutually disjoint [recall that by Lemma 5, the points from the set Cr(α) are c₀(AΔ)⁻¹-separated with a positive constant c₀].

1. Stable loops

These are connected components of Z(f) and $Z (\tilde{f})$ that do not intersect the set

U = U (Cr (α), δ) \overset{def}{=} (⋃_{p \in {Cr}_{E} (α)} D (p, δ)) ⋃ (⋃_{p \in {Cr}_{S} (α)} J (p, δ)) .

We denote by N_I(f) the number of stable loops in Z(f) and by $N_{I} (\tilde{f})$ the number of stable loops in $Z (\tilde{f})$ ⁠.

Observe that $D (p, \sqrt{\frac{a}{b}} δ) \subset J (p, δ)$ ⁠, p ∈ Cr_S(α), and that $\sqrt{\frac{a}{b}} ⩾ {(A Δ)}^{- 1 / 2}$ ⁠. We see that, for each p ∈ Cr(α), we have D(p, 2δ²) ⊂ U. Therefore, Lemma 11 applies to stable loops in Z(f) as well as to stable loops in $Z (\tilde{f})$ and yields a one-to-one correspondence between the set of stable loops in Z(f) and the set of stable loops in $Z (\tilde{f})$ ⁠. That is, $N_{I} (\tilde{f}) = N_{I} (f)$ ⁠.

2. Blinking circles

These are small connected components of $\tilde{f}$ that surround the points p ∈ Cr_E(α) and lie in the interiors of the corresponding disks D(p, δ). Recall that, by Lemma 6, $Z (\tilde{f})$ cannot intersect the boundary circle ∂D(p, δ) of such a disk.

By the summary in the end of the local extrema Sec. VI D 1, the number of such components is

N_{II} (\tilde{f}) \overset{def}{=} |\{p \in {Cr}_{E} (α) : \tilde{f} (p) and the eigenvalues of H_{f} (p) have opposite signs\}| .

3. The Bogomolny–Schmit loops

This is the most interesting part of $Z (\tilde{f})$ ⁠. Consider the graph G = G(f) embedded in $S^{2} (L)$ ⁠. The vertices of G are the joints J(p, δ),p ∈ Cr_S(α). The edges are connected components of the set

Z (f) \ ⋃_{p \in {Cr}_{S} (α)} J (p, δ)

(6.4)

that touch the boundaries ∂J(p, δ) [these components are homeomorphic to intervals, while the other connected components of the set (6.4) are homeomorphic to circles]. Each vertex of this graph has degree 4. The signs of $\tilde{f} (p)$ ⁠, p ∈ Cr_S(α), determine the way the graph G is turned into a collection of loops [(Fig. 3) see the summary in the end of the saddle point, Sec. VI D 2]. By $N_{III} (\tilde{f})$ ⁠, we denote the number of loops in this collection.

FIG. 3.

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Creation of the Bogomolny–Schmit loops.

4. The main lemma of Sec. VI

At last, we are able to state the main result of this section.

Lemma 12.

Let f ∈ C³(A, Δ, α, β), g ∈ C³(A), and

\tilde{f} = f + α^{'} g

. Suppose that the parameters

α^{'} ≪ α ≪ β ≪ 1 ≪ A ≪ Δ

(6.5)

satisfy following relations

A α^{'} ≪ α, A Δ^{2} β^{2} ≪ α ≪ {(A Δ)}^{- 2} β, A^{2} Δ^{3} α ≪ 1

(6.6)

and that

\min_{Cr (α)} | \tilde{f} | ≳ A Δ^{2} α^{2} .

(6.7)

Then,

N (\tilde{f}) = N_{I} (\tilde{f}) + N_{II} (\tilde{f}) + N_{III} (\tilde{f}) .

VII. LOWER BOUNDS FOR THE HESSIAN OF f_L ON THE ALMOST SINGULAR SET

Now, we return to regular Gaussian ensembles (f_L).

Lemma 13.

Given a sufficiently small positive ɛ, let α ⩽ L^−2+2ɛ and β²L^3ɛ ⩽ α. Then, there exists L₀ = L₀(ɛ) such that, for each L ⩾ L₀, w.h.p.,

\max_{Cr (α, β)} ‖ {(\nabla^{2} f_{L})}^{- 1} ‖_{op} ⩽ L^{3 ε},

where ‖⋅‖_op denotes the operator norm.

Proof.

Fix a small ɛ > 0 and consider the set Cr(5α, 4β). Let p be the probability that a given point

x \in S^{2} (L)

belongs to the set Cr(5α, 4β). By the invariance of the ensemble (f_L), this probability does not depend on x. The statistical independence of f_L(x) and ∇f_L(x) (Lemma 1), non-degeneracy of their distributions (Lemma 2), and uniform boundedness of their variances yield that

p = P \{| f_{L} (x) | ⩽ 5 α\} \cdot P \{| \nabla f_{L} (x) | ⩽ 4 β\} ≃ α β^{2} .

Next, note that, by Fubini’s theorem,

E [area (Cr (5 α, 4 β))] = p \cdot area (S^{2} (L)) ≃ p \cdot L^{2},

whence, by Chebyshev’s inequality,

P \{area (Cr (5 α, 4 β)) > p \cdot L^{2} \cdot L^{\frac{1}{2} ε}\} ≲ L^{- \frac{1}{2} ε} .

Thus, w.h.p.,

area (Cr (5 α, 4 β)) ⩽ p \cdot L^{2} \cdot L^{\frac{1}{2} ε} ≃ (α L^{2}) β^{2} L^{\frac{1}{2} ε} \overset{α L^{2} ⩽ L^{2 ε}}{⩽} β^{2} L^{2.5 ε} .

Denote by μ = μ(x) the eigenvalue of the Hessian matrix ∇²f_L with the minimal absolute value and by w = w(x) the corresponding normalized eigenvector. Assume that $‖ f_{L} ‖_{C^{3}} < \log L$ (recall that this holds w.o.p.), and suppose that, for some x ∈ Cr(α, β) and L ⩾ L₀, |μ(x)| < Δ⁻¹, where Δ = L^3ɛ. We will show that then, the set Cr(5α, 4β) contains a subset $\tilde{G} = \tilde{G} (x)$ with $area (\tilde{G}) ≳ β^{2} L^{3 ε} {(\log L)}^{- 1}$ ⁠. This will immediately imply the lemma.

Fix a point x ∈ Cr(α, β) with |μ(x)| < Δ⁻¹, take the corresponding map Ψ_x, and let F = f_L◦Ψ_x. Put τ = βΔ ≪ 1. For Y = tw(x), 0 ⩽ t ⩽ τ, we have

\begin{array}{l} | \nabla F (Y) | ⩽ | \nabla F (0) | + | \nabla^{2} F (0) Y | + O (| Y |^{2} ‖ F ‖_{C^{3} ({| X | ⩽ 1})}) \\ ⩽ β + Δ^{- 1} τ + τ^{2} O (\log L) = 2 β + β^{2} Δ^{2} O (\log L) ⩽ 3 β . \end{array}

Then, letting I = [0, tw(x)], we get

| F (Y) | ⩽ | F (0) | + \max_{I} | \nabla F | \cdot | Y | ⩽ α + 3 β τ = α + 3 β^{2} Δ ⩽ 4 α,

since β²Δ ⩽ α.

Put ρ = cβ(log L)⁻¹ with a sufficiently small positive constant c, and denote by Ω the ρ-neighborhood of the segment I on the plane Π_x. Then,

\max_{\bar{Ω}} | \nabla F | ⩽ \max_{I} | \nabla F | + O (ρ ‖ F ‖_{C^{3} ({| X | ⩽ 1})}) ⩽ 3 β + ρ O (\log L) < 3.5 β,

and

\begin{array}{l} \max_{\bar{Ω}} | F | ⩽ \max_{I} | F | + ρ \max_{I} | \nabla F | & + O (ρ^{2} ‖ F ‖_{C^{3} ({| X | ⩽ 1})}) \\ ⩽ 4 α + 3 β ρ + ρ^{2} O (\log L) = 4 α + β^{2} o (1) < 5 α . \end{array}

Let

\tilde{Ω} = Ψ_{x} (Ω)

⁠. We see that

\tilde{Ω} \subset Cr (5 α, 4 β)

⁠.

At the same time,

area (\tilde{Ω}) ≳ τ ρ = c β^{2} Δ {(\log L)}^{- 1} = c β^{2} L^{3 ε} {(\log L)}^{- 1},

completing the proof.□

The next lemma gives us a lower bound for the probability that a given point $x \in S^{2} (L)$ belongs to the set

Cr (α, β, Δ) = {x \in Cr (α, β) : ‖ {(\nabla^{2} f_{L} (x))}^{- 1} ‖_{op} ⩽ Δ} .

Let p be the probability that a given point $x \in S^{2} (L)$ belongs to the set Cr(α, β). By the invariance of the ensemble (f_L), this probability does not depend on x.

Lemma 14.

For any α, β ⩽ 1 and any Δ ⩾ 2,

P \{x \in Cr (α, β, Δ)\} ⩾ (1 - C {(\log Δ)}^{\frac{1}{4}} Δ^{- \frac{1}{2}}) p .

Proof.

We need to show that, conditioned on x ∈ Cr(α, β), the probability that

‖ {(\nabla^{2} f_{L} (x))}^{- 1} ‖_{op} ⩾ Δ

is

≲ {(\log Δ)}^{\frac{1}{4}} Δ^{- \frac{1}{2}}

⁠. Since f_L(x) and its Hessian ∇²f_L(x) are independent of the gradient ∇f_L(x), it will suffice to show that

P \{‖ {(\nabla^{2} f_{L} (x))}^{- 1} ‖_{op} > Δ |) | f (x) | ⩽ α\} ≲ {(\log Δ)}^{\frac{1}{4}} Δ^{- \frac{1}{2}} .

Fix $x \in S^{2} (L)$ and denote by μ₁(x) and μ₂(x) the eigenvalues of the Hessian. First, we show that, conditioned on the event {|f(x)| ⩽ α}, with large probability, |μ₁(x) ⋅ μ₂(x)| = |det ∇²f_L(x)| cannot be too small, and then that, with large probability, max(|μ₁(x)|, |μ₂(x)|) = ‖∇²f_L(x)‖_op cannot be too big. Together, these two estimates will do the job.

Fix coordinates (X₁, X₂) in the plane Π_x. Then,

\det \nabla^{2} f_{L} (x) = \partial_{1, 1}^{2} f_{L} (x) \partial_{2, 2}^{2} f_{L} (x) - {(\partial_{1, 2}^{2} f_{L} (x))}^{2} .

Recalling that, by Lemma 1,

\partial_{1, 2}^{2} f_{L} (x)

is independent of the vector

{(f_{L} (x), \partial_{1, 1}^{2} f_{L} (x), \partial_{2, 2}^{2} f_{L} (x))}^{t}

and that by Lemma 2, the distribution of

\partial_{1, 2}^{2} f_{L} (x)

does not degenerate, we conclude that, for any δ > 0,

P \{| μ_{1} (x) \cdot μ_{2} (x) | < δ |) | f (x) | ⩽ α\} ⩽ \sup_{s \in R} P \{| {(\partial_{1, 2}^{2} f_{L} (x))}^{2} - s | < δ\} ≲ \sqrt{δ} .

In the second step, taking into account that

‖ \nabla^{2} f_{L} (x) ‖_{op} ⩽ 2 \max_{1 ⩽ i, j ⩽ 2} | \partial_{i, j}^{2} f_{L} (x) |

⁠, we need to estimate from above the absolute values of the second order derivatives of f_L at x conditioned on f_L(x). The mixed derivative

\partial_{1, 2}^{2} f_{L} (x)

has a bounded variance and is independent of f_L(x). Furthermore, we use the normal correlation theorem, which says that if (θ, ξ) is a two-dimensional Gaussian vector, then the expectation and variance of θ conditioned on ξ equal

E [θ | ξ] = \frac{E [θ ξ]}{Var [ξ]} ξ

and

Var [θ | ξ] = Var [θ] - \frac{{(E [θ ξ])}^{2}}{Var [ξ]} ⩽ Var [θ] .

By this theorem, the distribution of

\partial_{i, i}^{2} f_{L} (x)

⁠, i = 1, 2, conditioned on f_L(x), is normal with bounded conditional mean

|E [\partial_{i, i}^{2} f_{L} (x) | f_{L} (x)]| = |E [f_{L} (x) \partial_{i, i}^{2} f_{L} (x)]| \cdot |f_{L} (x)| ≲ 1

(recall that f_L is normal and that we are interested only in the values |f_L(x)| ⩽ α ⩽ 1) and with the bounded conditional variance

Var [\partial_{i, i}^{2} f_{L} (x) | f_{L} (x)] ⩽ Var [\partial_{i, i}^{2} f_{L} (x)] ≲ 1 .

Thus,

P \{\max (| μ_{1} (x) |, | μ_{2} (x) |) > λ |) | f (x) | ⩽ α\} ≲ e^{- c λ^{2}}

⁠. Therefore, after conditioning on {|f(x)| ⩽ α}, with probability at least

1 - C (\sqrt{δ} + e^{- c λ^{2}})

⁠, we have

\begin{array}{l} δ ⩽ | μ_{1} (x) | \cdot | μ_{2} (x) | & = \min (| μ_{1} (x) |, | μ_{2} (x) |) \cdot \max (| μ_{1} (x) |, | μ_{2} (x) |) \\ ⩽ λ \min (| μ_{1} (x) |, | μ_{2} (x) |) = λ ‖ {(\nabla^{2} f_{L} (x))}^{- 1} ‖_{op}^{- 1} . \end{array}

Letting λ = δΔ and

δ = \frac{1}{\sqrt{2 c}} Δ^{- 1} \sqrt{\log Δ}

and noting that

\sqrt{δ} + e^{- c λ^{2}} ≲ Δ^{- \frac{1}{2}} {(\log Δ)}^{\frac{1}{4}}

⁠, we complete the proof.□

VIII. THE TWO-POINT FUNCTION OF THE SET Cr(α, β)

In this section, we will look at the set Cr(α, β) of almost singular points of f_L,

Cr (α, β) = \{x \in S^{2} (L) : | f_{L} (x) | ⩽ α, | \nabla f_{L} (x) | ⩽ β\},

with very small parameters α and β, and at its one- and two-point functions p and p(x, y). As above, p is the probability that a given point $x \in S^{2} (L)$ belongs to the set Cr(α, β). Recall that, by the invariance of the ensemble (f_L), this probability does not depend on x and that the statistical independence of f_L(x) and ∇f_L(x) yields that

p = P \{| f_{L} (x) | ⩽ α\} \cdot P \{| \nabla f_{L} (x) | ⩽ β\} ≃ α β^{2} .

By p(x, y), we denote the probability that two given points $x, y \in S^{2} (L)$ belong to the set Cr(α, β). By the invariance of the distribution of f_L with respect to isometries of the sphere, p(x, y) depends only on the spherical distance between the points x and y.

A. Estimates of the two-point function

Lemma 15.

Let (f_L) be a regular Gaussian ensemble. Then, the estimates hold uniformly in α, β ⩽ 1 and in L ⩾ L₀,

p (x, y) ≲ \max \{d_{L} {(x, y)}^{- Θ}, 1\} p^{2},

(8.1)

with some positive constant Θ, and for d_L(x, y) ⩾ 1, we have p(x, y) = Wp², with

| W - 1 | ≲ d_{L} {(x, y)}^{- γ} .

(8.2)

Note that the proof of the short-distance estimate (8.1) is quite lengthy, while the long-distance estimate (8.2) is a straightforward consequence of the power decay of correlations of (f_L).

B. Proof of the short-distance estimate (8.1)

1. Beginning the proof

Let $x, y \in S^{2} (L)$ ⁠. Fix the coordinate systems in the planes Π_x and Π_y, and let Γ(x, y) be the covariance matrix of the Gaussian six-dimensional vector,

v (x, y) = {(f_{L} (x), \nabla f_{L} (x), f_{L} (y), \nabla f_{L} (y))}^{t} .

Then,

p (x, y) = \frac{1}{{(2 π)}^{3} \sqrt{\det Γ (x, y)}} \int_{Ω} \exp [- \frac{1}{2} ξ^{t} Γ {(x, y)}^{- 1} ξ] d^{6} ξ,

where

Ω = \{ξ \in R^{6} : | ξ_{1} |, | ξ_{4} | ⩽ α, | ξ_{2} |^{2} + | ξ_{3} |^{2}, | ξ_{5} |^{2} + | ξ_{6} |^{2} ⩽ β^{2}\} .

Note that vol(Ω) ≲ α²β⁴ ≃ p². Hence, to prove estimate (8.1), we need to bound from below the minimal eigenvalue λ = λ(x, y) of the covariance matrix Γ = Γ(x, y),

λ = \min \{ξ^{t} Γ (x, y) ξ : ξ \in R^{6}, | ξ | = 1\} .

Note that λ does not depend on the choice of the coordinate systems in the planes Π_x and Π_y.

First, we show that there exists a sufficiently large constant d₀, independent of L, so that λ(x, y) is bounded from below by a positive constant whenever d_L(x, y) ⩾ d₀. Hence, proving estimate (8.1), we assume that d_L(x, y) ⩽ d₀ [while later, proving the long-distance estimate (8.2), we will assume that d_L(x, y) ⩾ d₀]. The value of sufficiently large constant d₀ is inessential for our purposes.

Denote by $C$ the covariance matrix of ∇f_L(x) and put

\tilde{Γ} = (\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & C & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & C \end{matrix}) .

Since the matrix $C$ is non-degenerate uniformly in L ⩾ L₀ (Lemma 2), the matrix $\tilde{Γ}$ is also non-degenerate uniformly in L ⩾ L₀.

By our assumption on the power decay of correlations, we have

\max_{1 ⩽ i, j ⩽ 6} |Γ_{i j} (x, y) - {\tilde{Γ}}_{i j}| ≲ {(1 + d_{L} (x, y))}^{- γ} .

Then, provided that d_L(x, y) ⩾ d₀ with d₀ ≫ 1, we get

‖ Γ {(x, y)}^{- 1} - {\tilde{Γ}}^{- 1} ‖_{op} ≲ d_{L} {(x, y)}^{- γ},

and therefore,

‖ Γ {(x, y)}^{- 1} ‖_{op} ⩾ ‖ {\tilde{Γ}}^{- 1} ‖_{op} - O (d_{L} {(x, y)}^{- γ}) ⩾ \frac{1}{2} ‖ {\tilde{Γ}}^{- 1} ‖_{op} .

Thus, until the end of the proof of the short-distance estimate (8.1), we assume that d_L(x, y) ⩽ d₀ with some positive d₀ independent of L.

Let v(x) denote the three-dimensional Gaussian vector $v (x) = {(f_{L} (x), \nabla f_{L} (x))}^{t}$ ⁠, and let $a = {(ξ_{1}, ξ_{2}, ξ_{3})}^{t}$ and $b = {(ξ_{4}, ξ_{5}, ξ_{6})}^{t}$ ⁠. Then,

ξ^{t} Γ (x, y) ξ = E [{⟨ v (x, y), ξ ⟩}^{2}] = E [{(⟨ v (x), a ⟩ + ⟨ v (y), b ⟩)}^{2}],

whence

λ = \min \{E [{(〈 v (x), a 〉 + 〈 v (y), b 〉)}^{2}] : a, b \in R^{3}, | a |^{2} + | b |^{2} = 1\} .

By the compactness of the unit sphere in $R^{3}$ ⁠, there exist $a, b \in R^{3}$ ⁠, |a|² + |b|² = 1, such that

λ = E [{(〈 v (x), a 〉 + 〈 v (y), b 〉)}^{2}] = ‖ 〈 v (x), a 〉 + 〈 v (y), b 〉 ‖^{2} .

(8.3)

Here and until the end of the Proof of Lemma 15, ‖⋅‖ stands for the L²-norm, i.e., $‖ η ‖^{2} = E [| η |^{2}] = Var [η]$ for the Gaussian random variable η.

2. The normal and the tangential derivatives

Let $C$ be the big circle on $S^{2} (L)$ that passes through the points x and y, and let $I \subset C$ be the shortest of the two arcs of $C$ with the endpoints x and y. We orient $C$ by moving from x to y along $I$ and choose the coordinate systems in Π_x and Π_y so that one coordinate vector is parallel to the tangent to $C$ (at x and y correspondingly), while the other one is orthogonal to $C$ ⁠. We keep the same orientation for both coordinate systems. We denote by ∂_‖ the derivative along $C$ and by ∂_⊥ the derivative in the normal direction to $C$ and decompose

v (x) = v_{‖} (x) + v_{⊥} (x),

where $v_{‖} (x) = {(f_{L} (x), \partial_{‖} f_{L} (x), 0)}^{t}$ and $v_{⊥} (x) = {(0, 0, \partial_{⊥} v (x))}^{t}$ ⁠. Then, by (8.3),

λ = ‖ 〈 v_{‖} (x), a^{'} 〉 + 〈 v_{⊥} (x), a^{''} 〉 + 〈 v_{‖} (y), b^{'} 〉 + 〈 v_{⊥} (y), b^{''} 〉 ‖^{2},

(8.4)

where $a^{'} = {(a_{1}, a_{2}, 0)}^{t}$ and $a^{''} = {(0, 0, a_{3})}^{t}$ ⁠, similarly for b′ and b^″, and |a′|² + |a^″|² + |b′|² + |b^″|² = 1.

Now, consider another Gaussian six-dimensional vector

\tilde{v} (x, y) = {(f_{L} (x), \partial_{‖} f_{L} (x), - \partial_{⊥} f_{L} (x), f_{L} (y), \partial_{‖} f_{L} (y), - \partial_{⊥} f_{L} (y))}^{t} .

Since the distribution of f_L is invariant with respect to orthogonal transformations, the Gaussian vectors v(x, y) and $\tilde{v} (x, y)$ have the same covariance matrix in the chosen coordinate systems in Π_x and Π_y. Therefore,

λ = ‖ 〈 v_{‖} (x), a^{'} 〉 - 〈 v_{⊥} (x), a^{''} 〉 + 〈 v_{‖} (y), b^{'} 〉 - 〈 v_{⊥} (y), b^{''} 〉 ‖^{2} .

(8.5)

Juxtaposing (8.4) with (8.5), we conclude that

λ = ‖ 〈 v_{‖} (x), a^{'} 〉 + 〈 v_{‖} (y), b^{'} 〉 ‖^{2} + ‖ 〈 v_{⊥} (x), a^{''} 〉 + 〈 v_{⊥} (y), b^{''} 〉 ‖^{2} .

We split the rest of the proof of estimate (8.1) into two cases: (i) $| a^{''} |^{2} + | b^{''} |^{2} ⩾ \frac{1}{2}$ and (ii) $| a^{'} |^{2} + | b^{'} |^{2} ⩾ \frac{1}{2}$ ⁠. In the first case, we shall use a straightforward argument, while in the second case, we shall prove an equivalent estimate on the Fourier side.

3. Case (i): $| a^{''} |^{2} + | b^{''} |^{2} ⩾ \frac{1}{2}$

In this case, we use the estimate λ ≳‖⟨v_⊥(x), a^″⟩ + ⟨v_⊥(y), b^″⟩‖². By the invariance of the distribution of f_L, the random pairs (∂_⊥f_L(x), ∂_⊥f_L(y)) and (∂_⊥f_L(y), ∂_⊥f_L(x)) have the same distribution. Therefore,

‖ 〈 v_{⊥} (x), a^{''} 〉 + 〈 v_{⊥} (y), b^{''} 〉 ‖ = ‖ 〈 v_{⊥} (y), a^{''} 〉 + 〈 v_{⊥} (x), b^{''} 〉 ‖ .

Since $| a^{''} |^{2} + | b^{''} |^{2} ⩾ \frac{1}{2}$ ⁠, at least one of the following holds:

either $| a^{''} + b^{''} | ⩾ \frac{1}{2}$ or $| a^{''} - b^{''} | ⩾ \frac{1}{2}$ ⁠.

We assume that $| a^{''} + b^{''} | ⩾ \frac{1}{2}$ (the other case is similar and slightly simpler), let e = a^″ + b^″, $| e | ⩾ \frac{1}{2}$ ⁠, and notice that

\begin{array}{l} ‖ ⟨ v_{⊥} (x) + v_{⊥} (y), e ⟩ ‖ & = ‖ (⟨ v_{⊥} (x), a^{″} ⟩ + ⟨ v_{⊥} (y), b^{″} ⟩) + (⟨ v_{⊥} (x), b^{″} ⟩ + ⟨ v_{⊥} (y), a^{″} ⟩) ‖ \\ ⩽ ‖ ⟨ v_{⊥} (x), a^{″} ⟩ + ⟨ v_{⊥} (y), b^{″} ⟩ ‖ + ‖ ⟨ v_{⊥} (x), b^{″} ⟩ + ⟨ v_{⊥} (y), a^{″} ⟩ ‖ ≲ \sqrt{λ} . \end{array}

We take the point $z \in C$ ⁠, z ≠ x, so that d_L(y, z) = d_L(x, y). Then, by the invariance of the distribution of f_L with respect to the isometries of the sphere,

‖ 〈 v_{⊥} (y) + v_{⊥} (z), e 〉 ‖ = ‖ 〈 v_{⊥} (x) + v_{⊥} (y), e 〉 ‖,

whence

\sqrt{λ} ≳ ‖ ⟨ v_{⊥} (x) - v_{⊥} (z), e ⟩ ‖ .

Put x₀ = x and x₁ = z, then take the point $x_{2} \in C$ ⁠, x₂≠x₀, so that d_L(x₂, x₁) = d_L(x₁, x₀), and continue this way until d_L(x₀, x_N) ⩾ d₀, where d₀ is the correlation length defined above. Then,

‖ ⟨ v_{⊥} (x_{0}) - v_{⊥} (x_{N}), e ⟩ ‖ ≲ N \sqrt{λ} .

On the other hand, since d₀ is the correlation length and d_L(x₀, x_N) ⩾ d₀, we have

‖ ⟨ v_{⊥} (x_{0}) - v_{⊥} (x_{N}), e ⟩ ‖^{2} ≳ ‖ ⟨ v_{⊥} (x_{0}), e ⟩ ‖^{2} + ‖ ⟨ v_{⊥} (x_{N}), e ⟩ ‖^{2} ≳ 1

(at the last step, we use that the distribution of v_⊥ does not degenerate uniformly in L ⩾ L₀ and that $| e | ⩾ \frac{1}{2}$ ⁠). Therefore, λ ≳ N⁻².

Recalling that by the definition of N, we have (N − 1)d_L(x, y) < d₀, and we get λ ≳ d_L(x,y)², which concludes our consideration of the first case.

4. Case (ii): $| a^{'} |^{2} + | b^{'} |^{2} ⩾ \frac{1}{2}$

In this case, we restrict the function f_L to the big circle $C$ and treat it as a periodic random Gaussian function $F : R \to R$ with translation-invariant distribution. To simplify the notation, we omit the index L. By ρ, we denote the spectral measure of F, that is,

E [F (X) F (Y)] = \hat{ρ} (X - Y) = \int_{R} e^{2 π i ξ (X - Y)} d ρ (ξ),

where $X, Y \in R$ correspond to the points $x, y \in C$ ⁠. Then, we have

\begin{array}{l} λ & ≳ E [{(〈 v_{‖} (X), a^{'} 〉 + 〈 v_{‖} (Y), b^{'} 〉)}^{2}] \\ = \int_{R} {|(a_{1} + a_{2} ξ) e^{2 π i ξ X} + (b_{1} + b_{2} ξ) e^{2 π i ξ Y}|}^{2} d ρ (ξ) \\ = \int_{R} {|(a_{1} + a_{2} ξ) + (b_{1} + b_{2} ξ) e^{i δ ξ}|}^{2} d ρ (ξ), \end{array}

where $δ = \frac{1}{2 π} | X - Y | = \frac{1}{2 π} d_{L} (x, y)$ ⁠. Furthermore,

ρ (R) = E [F {(0)}^{2}] = 1,

m \overset{def}{=} \int_{R} ξ^{2} d ρ (ξ) = E [F^{'} {(0)}^{2}], m ≃ 1,

by the uniform non-degeneracy of ∇f_L and

| \hat{ρ} (s) | = | E [F (0) F (s)] | ≲ | s |^{- γ}, | s | ⩽ \frac{1}{2} L,

by the power decay of correlations of f_L. Note that since the function F is 2πL-periodic, the Fourier transform $\hat{ρ}$ of its spectral measure is also 2πL-periodic.

Recalling that $\frac{1}{2} ⩽ | a^{'} |^{2} + | b^{'} |^{2} ⩽ 1$ ⁠, we notice that $| a_{1} | + | a_{2} | + | b_{1} | + | b_{2} | ⩾ \frac{1}{2}$ as well. These remarks reduce the lower bound for λ we are after to a question in harmonic analysis.

Given δ > 0, consider the exponential sum of degree 4,

p_{δ} (ξ) = (a_{1} + a_{2} ξ) + (b_{1} + b_{2} ξ) e^{i δ ξ}, a_{1}, a_{2}, b_{1}, b_{2} \in C,

(8.6)

and denote ‖p_δ‖_W = |a₁| + |a₂| + |b₁| + |b₂| (recall that an exponential sum is the expression

S (ξ) = \sum_{j = 1}^{n} q_{j} (ξ) e^{i λ_{j} ξ}, \deg S \overset{def}{=} \sum_{j = 1}^{n} (\deg q_{j} + 1),

where λ_j are real numbers and q_j are polynomials in ξ with complex coefficients). Then, the following lemma does the job.

Lemma 16.

Let p_δ be the exponential sum () of degree 4. Let ρ be a probability measure on

R

with the 2πL-periodic Fourier transform

\hat{ρ}

. Assume that

m = \int_{R} ξ^{2} d ρ (ξ) ≃ 1

and

| \hat{ρ} (s) | ⩽ C | s |^{- γ}, | s | ⩽ \frac{1}{2} L .

Then, given δ₀ > 0, there exist c = c(C, γ, m, δ₀) > 0 and L₀ = L₀(C, γ, m, δ₀) such that, for every 0 < δ ⩽ δ₀ and every L ⩾ L₀,

\int_{R} | p_{δ} |^{2} d ρ ⩾ c δ^{6} ‖ p_{δ} ‖_{W}^{2} .

C. Proof of Lemma 16

1. Beginning the Proof of Lemma 16

Let $A = 2 \sqrt{m}$ ⁠. Then, by Chebyshev’s inequality,

ρ (R \ [- A, A]) ⩽ A^{- 2} m = \frac{1}{4} .

Take B = δ₀A and let 0 < δ ⩽ δ₀. Then, [−A, A] ⊂ [−Bδ⁻¹, Bδ⁻¹]. Let κ > 0 be a sufficiently small parameter, which we will choose later, and consider the set

Ξ = \{ξ \in [- B δ^{- 1}, B δ^{- 1}] : | p_{δ} (ξ) | < ‖ p_{δ} ‖_{W} κ^{3} δ^{3}\} .

We claim that

the set Ξ is a union of at most B + 5 intervals I_j, 1 ⩽ j ⩽ B + 5, and
the length of each interval I_j is ⩽ C(B, δ₀) κ.

Having these claims, we will show that $ρ ([- A, A] \ Ξ) ⩾ \frac{1}{4}$ ⁠, whence

\int_{[- A, A] \ Ξ} | p_{δ} |^{2} d ρ ⩾ \frac{1}{4} {(κ δ)}^{6} ‖ p_{δ} ‖_{W}^{2} .

This will complete the Proof of Lemma 16.

2. Proof of Claim (a)

To show (a), we consider the exponential sum P = |p_δ|² of degree 9. By the classical Langer lemma (see Lemma 1.3 of Ref. 3), the number of zeroes of any exponential sum of degree N on any interval $J \subset R$ cannot exceed

(N - 1) + \frac{Δ}{2 π} | J |,

where Δ is the maximal distance between the exponents in the exponential sum. Hence, the number of solutions to equation P(ξ) = t on the interval [−Bδ⁻¹, Bδ⁻¹] does not exceed

8 + \frac{4 δ}{2 π} \cdot 2 B δ^{- 1} < 8 + 2 B .

Hence, the set Ξ consists of at most $\frac{1}{2} ((8 + 2 B) + 2) = 5 + B$ intervals, proving (a).

3. Proof of Claim (b)

To show (b), we apply Turán’s lemma (see Theorem 1.5 of Ref. 3), which states that for any exponential sum S of degree N and any pair of closed intervals I ⊂ J,

\max_{J} | S | ⩽ {(\frac{C | J |}{| I |})}^{N - 1} \max_{I} | S |,

where C is a numerical constant. Applying this lemma to the exponential sum p_δ of degree 4 and to each of the intervals I_j ⊂ [−Bδ⁻¹, Bδ⁻¹], we get

\begin{array}{l} \max_{[- B δ^{- 1}, B δ^{- 1}]} | p_{δ} | & ⩽ {(\frac{C \cdot 2 B δ^{- 1}}{| I_{j} |})}^{3} \max_{I_{j}} | p_{δ} | \\ ⩽ {(\frac{C \cdot 2 B δ^{- 1}}{| I_{j} |})}^{3} ‖ p_{δ} ‖_{W} κ^{3} δ^{3} (since I_{j} \subset Ξ) \\ = κ^{3} {(\frac{C \cdot 2 B}{| I_{j} |})}^{3} ‖ p_{δ} ‖_{W} . \end{array}

On the other hand,

\begin{array}{l} \max_{[- B δ^{- 1}, B δ^{- 1}]} | p_{δ} | & = \max_{ξ \in [- B, B]} |(a_{1} + δ^{- 1} a_{2} ξ) + (b_{1} + δ^{- 1} b_{2} ξ) e^{i ξ}| (by scaling) \\ ⩾ c (B) (| a_{1} | + δ^{- 1} | a_{2} | + | b_{1} | + δ^{- 1} | b_{2} |) (by compactness) \\ \overset{δ ⩽ δ_{0}}{⩾} c (B) {(1 + δ_{0})}^{- 1} ‖ p_{δ} ‖_{W} . \end{array}

Thus, |I_j| ⩽ C(B, δ₀)κ, proving (b).

4. Completing the Proof of Lemma 16

Recall that $ρ ([- A, A]) ⩾ \frac{3}{4}$ and that given κ > 0, we defined the set

Ξ = \{ξ \in [- B δ^{- 1}, B δ^{- 1}] : | p_{δ} (ξ) | < ‖ p_{δ} ‖_{W} κ^{3} δ^{3}\},

satisfying (a) and (b). Then, we have the following alternative:

either $ρ ([- A, A] \ Ξ) ⩾ \frac{1}{4}$ or $ρ ([- A, A] \cap Ξ) ⩾ \frac{1}{2}$ ⁠.

In the first case,

\int_{R} | p_{δ} |^{2} d ρ ⩾ \int_{[- A, A] \ Ξ} | p_{δ} |^{2} d ρ ⩾ \frac{1}{4} ‖ p_{δ} ‖_{W}^{2} κ^{6} δ^{6},

and we are done (modulo the choice of the parameter κ, which will be made later). It remains to show that ifκis sufficiently small, then the second case cannot occur.

Suppose that $ρ ([- A, A] \cap Ξ) ⩾ \frac{1}{2}$ ⁠. Then, $ρ (Ξ) ⩾ \frac{1}{2}$ ⁠. By claim (a), Ξ is a union of at most B + 5 intervals I_j, hence, for at least one of them, ρ(I_j) ⩾ c(B) > 0. We call this interval I and denote by ν the restriction of the measure ρ on I. We choose a large parameter S so that 1 ≪ S ≪ κ⁻¹ and estimate the integral

J = \int_{- S}^{S} (1 - \frac{| s |}{S}) {|\hat{ν} (s)|}^{2} d s

from below and from above, obtaining the estimates that will contradict each other.

Denote by ξ_I the center of the interval I. Then, for |s| ⩽ S and ξ ∈ I, we have

\begin{array}{l} |e^{2 π i ξ s} - e^{2 π i ξ_{I} s}| & ⩽ 2 π | s | \cdot | ξ - ξ_{I} | \\ ⩽ 2 π S \cdot \frac{1}{2} | I | \\ ⩽ 2 π S \cdot κ \cdot \frac{1}{2} C (B, δ_{0}) (since, by claim (b), | I | ⩽ C (B, δ_{0}) κ) \\ < \frac{1}{2} \end{array}

provided that S ⋅ κ is sufficiently small. Therefore, for |s| ⩽ S, we have

{|\hat{ν} (s)|}^{2} = {|\int_{I} e^{i s ξ} d ρ (ξ)|}^{2} ⩾ \frac{1}{4} | ρ (I) |^{2} ⩾ c_{1} (B),

whence

J ⩾ c_{1} (B) \int_{- S}^{S} (1 - \frac{| s |}{S}) d s ⩾ c_{2} (B) S .

On the other hand, using the identity

\int_{R} φ (s) | \hat{ν} (s) |^{2} d s = \iint_{R \times R} \hat{φ} (λ - η) d ν (λ) d ν (η),

with φ(s) = (1 −|s|/S)₊ and noting that the Fourier transform of this function is non-negative, we get

\int_{- S}^{S} (1 - \frac{| s |}{S}) {|\hat{ν} (s)|}^{2} d s ⩽ \int_{- S}^{S} (1 - \frac{| s |}{S}) {|\hat{ρ} (s)|}^{2} d s .

Then, recalling that the function $\hat{ρ}$ is 2πL-periodic and that $| \hat{ρ} (s) | ⩽ \min (1, C | s |^{- γ})$ for |s| ⩽ L/2 and assuming without loss of generality that $γ < \frac{1}{2}$ ⁠, we get

J ≲ S \cdot {(\min (S, L))}^{- 2 γ} .

Choosing κ sufficiently small and S sufficiently large, we arrive at a contradiction, which completes the Proof of Lemma 16, and therefore, that of estimate (8.1) in Lemma 15.□

D. Proof of the long-distance estimate (8.2)

As above, we denote by $C$ the covariance matrix of ∇f_L(x) and put

\tilde{Γ} = (\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & C & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & C \end{matrix}) .

Since the matrix $C$ is non-degenerate uniformly in L ⩾ L₀, the matrix $\tilde{Γ}$ is also non-degenerate uniformly in L ⩾ L₀.

We assume that d_L(x, y) ⩾ d₀, where d₀ is sufficiently large (and independent of L). Then, we have

\max_{1 ⩽ i, j ⩽ 6} |Γ_{i j} (x, y) - {\tilde{Γ}}_{i j}| ≲ d_{L} {(x, y)}^{- γ} .

Therefore,

\det Γ (x, y) = \det \tilde{Γ} + O (d_{L} {(x, y)}^{- γ})

and

‖ Γ {(x, y)}^{- 1} - {\tilde{Γ}}^{- 1} ‖_{op} ≲ d_{L} {(x, y)}^{- γ} .

Recall that

p (x, y) = \frac{1}{{(2 π)}^{3} \sqrt{\det Γ (x, y)}} \int_{Ω} \exp [- \frac{1}{2} ξ^{t} Γ {(x, y)}^{- 1} ξ] d^{6} ξ,

where

Ω = \{ξ \in R^{6} : | ξ_{1} |, | ξ_{4} | ⩽ α, | ξ_{2} |^{2} + | ξ_{3} |^{2}, | ξ_{5} |^{2} + | ξ_{6} |^{2} ⩽ β^{2}\},

and that

\frac{1}{{(2 π)}^{3} \sqrt{\det \tilde{Γ}}} \int_{Ω} \exp [- \frac{1}{2} ξ^{t} {\tilde{Γ}}^{- 1} ξ] d^{6} ξ = p^{2} .

Hence,

\begin{array}{l} p (x, y) & = {(2 π)}^{- 3} () \det \tilde{Γ} + O {(d_{L} {(x, y)}^{- γ})}^{- \frac{1}{2}} \int_{Ω} \exp [- \frac{1}{2} ξ^{t} {\tilde{Γ}}^{- 1} ξ + O (d_{L} {(x, y)}^{- γ})] d^{6} ξ \\ = {(2 π)}^{- 3} () 1 + O (d_{L} {(x, y)}^{- γ}) \frac{1}{\sqrt{\det \tilde{Γ}}} \int_{Ω} \exp [- \frac{1}{2} ξ^{t} {\tilde{Γ}}^{- 1} ξ] d^{6} ξ \\ = () 1 + O (d_{L} {(x, y)}^{- γ}) p^{2}, \end{array}

completing the proof of estimate (8.2) in Lemma 15.□

IX. STRUCTURE OF THE SET Cr(α) WITH L^−2+ɛ ⩽ α ⩽ L^−2+2ɛ

Now, we are ready to prove our main lemma:

Lemma 17.

There exist positive ɛ₀ and c and positive C such that, given 0 < ɛ ⩽ ɛ₀ and L ⩾ L₀(ɛ), for L^−2+ɛ ⩽ α ⩽ L^−2+2ɛ, w.h.p., the set Cr(α) is L^1−Cɛ-separated, and |Cr(α)| ⩾ L^cɛ.

A. w.h.p., the set Cr(α) is L^1−Cɛ-separated

In this part, we assume that α ⩽ L^−2+2ɛ, choose β and ρ so that

β^{2} L^{3 ε} ⩽ α, ρ = β {(\log L)}^{- 1},

and fix a maximal ρ-separated set $X (ρ)$ on $S^{2} (L)$ ⁠. Then, $| X (ρ) | ≃ {(L / ρ)}^{2}$ ⁠.

First, we note that, w.h.p., the points of the set Cr(α) are L^−4ɛ-separated. This is a straightforward consequence of part (B) in Lemma 4 combined with a priori w.o.p.-bound $‖ f_{L} ‖_{C^{3}} < \log L$ and with the w.h.p.-estimate $\max_{Cr (α)} ‖ {(\nabla^{2} f_{L})}^{- 1} ‖_{op} ⩽ L^{3 ε}$ provided by Lemma 13. Hence, we need to estimate the probability of the event

E = \{\exists z_{1}, z_{2} \in Cr (α) : L^{- 4 ε} ⩽ d_{L} (z_{1}, z_{2}) ⩽ L^{1 - C ε}\}

with an appropriately chosen constant C.

Suppose that the event $E$ occurs. Denote by x₁ and x₂ the closest to z₁ and z₂ points in $X (ρ)$ ⁠. Then,

\frac{1}{2} L^{- 4 ε} ⩽ d_{L} (x_{1}, x_{2}) ⩽ 2 L^{1 - C ε},

(9.1)

and

\begin{array}{l} | f_{L} (x_{i}) | ⩽ α + O (ρ^{2} ‖ f_{L} ‖_{C^{2}}) \overset{w.o.p.}{<} α + O (β^{2} {(\log L)}^{- 1}) < 2 α, \\ | \nabla f_{L} (x_{i}) | ⩽ O (ρ ‖ f_{L} ‖_{C^{2}}) \overset{w.o.p.}{<} β {(\log L)}^{- 1} \cdot \log L = β, \end{array}

i.e., x₁, x₂ ∈ Cr(2α, β). We claim that

the mean number of pairs of points $x_{1}, x_{2} \in Cr (2 α, β) ⋂ X (ρ)$ satisfying (9.1) is bounded from above by L^−ɛ.

By Chebyshev’s inequality, this yields that the probability that there exists at least one such pair is also bounded from above by L^−ɛ, which proves the L^1−Cɛ-separation.

The mean we need to estimate equals

\sum_{\begin{array}{c} x_{1}, x_{2} \in Cr (2 α, β) \cap X (ρ) \\ (9.1) occurs \end{array}} p (x_{1}, x_{2}),

where $p (x_{1}, x_{2}) = P {x_{1}, x_{2} \in Cr (2 α, β)}$ is the two-point function estimated in Lemma 15. By Lemma 15, p(x₁, x₂) ≲ L^4ɛΘp², so the whole sum is

\begin{array}{l} ≲ \sum_{x_{1} \in X (ρ)} L^{4 ε Θ} p^{2} \cdot |{x_{2} \in X (r) : d_{L} (x_{1}, x_{2}) ⩽ 2 L^{1 - C ϵ}}| \\ ≲ L^{4 ε Θ} p^{2} \cdot {(L^{1 - C ε} / ρ)}^{2} \cdot {(L / ρ)}^{2} \\ ≲ L^{- (C - 4 Θ) ε} α^{2} β^{4} \cdot (L^{4} / β^{4}) \log^{4} L (p ≃ α β^{2}, ρ = β / \log L) \\ ≲ L^{- (C - 4 Θ) ε} \cdot L^{4 ε} \log^{4} L (α ⩽ L^{- 2 + 2 ε}) \\ < L^{- ε}, \end{array}

provided that the constant C is sufficiently large. This proves that the set Cr(α) is L^1−Cɛ-separated.□

B. w.h.p., |Cr(α)| ⩾ L^cɛ

In this part, we assume that α ⩾ L^−2+ɛ and introduce the parameters β, Δ, and r, satisfying

Δ = L^{\frac{1}{4} ε}, β^{2} L^{3 ε} = \frac{1}{3} α, r = β Δ .

We fix a maximal r-separated set $X (r)$ on $S^{2} (L)$ ⁠. Then, the disks D(x, r), $x \in X (r)$ ⁠, cover $S^{2} (L)$ with a bounded multiplicity of covering and $| X (r) | ≃ {(L / r)}^{2}$ ⁠. We set

Y = X (r) \cap Cr (\frac{1}{3} α, β, \frac{1}{2} Δ) .

W.o.p., for L ⩾ L₀, we have $‖ f_{L} ‖_{C^{3}} ⩽ \log L$ ⁠. Then, by Lemma 4, each disk D(y, r), y ∈ Y, contains a unique critical point z ∈ Cr(α), and

\begin{array}{l} ‖ {(\nabla^{2} f_{L} (z))}^{- 1} ‖_{op} ⩽ ‖ {(\nabla^{2} f_{L} (y))}^{- 1} ‖_{op} (1 + O (r ‖ {(\nabla^{2} f_{L} (y))}^{- 1} ‖_{op} \cdot ‖ f_{L} ‖_{C^{3}})) \\ ⩽ \frac{1}{2} Δ + O (r Δ^{2} \log L) < Δ, \end{array}

provided that L ⩾ L₀. This yields two useful observations that hold w.o.p.:

|Cr(α)|≳|Y|, and
if y₁, y₂ ∈ Y, then either d_L(y₁, y₂) ⩽ 2r and the number of such pairs (y₁, y₂) is $≲ | X (r) |$ or d(y₁, y₂) ⩾ L^−ɛ. Indeed, if the points y₁, y₂ ∈ Y generate the same critical point z, then d_L(y₁, y₂) ⩽ 2r. If they generate different critical points z, we note that, by part (B) of Lemma 4, the set of critical points z of f_L with $‖ {(\nabla^{2} f_{L} (z))}^{- 1} ‖_{op} ⩽ Δ$ is cΔ⁻¹(log L)⁻¹-separated, and thus, in this case, d(y₁, y₂) ⩾ L^−ɛ.

In the following, we will show that, for sufficiently large L,

E [| Y |] ≳ L^{\frac{1}{2} ε}

and that

Var [| Y |] ≲ L^{- c ε} {(E [| Y |])}^{2} .

These two estimates combined with the first observation readily yield what we need.

1. Estimating $E [| Y |]$

This estimate is straightforward,

E [| Y |] = P \{x \in Cr (\frac{1}{3} α, β, \frac{1}{2} Δ)\} \cdot | X (r) |

≳ P \{x \in Cr (\frac{1}{3} α, β)\} \cdot {(\frac{L}{r})}^{2} (by Lemma14)

≳ α β^{2} \cdot \frac{L^{2}}{β^{2} Δ^{2}}

(9.2)

⩾ L^{\frac{1}{2} ε} (α ⩾ L^{- 2 + ε}, Δ = L^{\frac{1}{4} ε}) .

(9.3)

2. Estimating Var[|Y|]

In this section, p and p(x, y) will denote the one- and two-point functions of the set $Cr (\frac{1}{3} α, β)$ ⁠. Given $x, y \in S^{2} (L)$ ⁠, we put

p_{Δ} = P \{y \in Cr (\frac{1}{3} α, β, \frac{1}{2} Δ)\}, p_{Δ} (x, y) = P \{x, y \in Cr (\frac{1}{3} α, β, \frac{1}{2} Δ)\} .

Then, $E [| Y |] = p_{Δ} | X (r) |$ and

Var [| Y |] = \sum_{y \in X (r)} (p_{Δ} - p_{Δ}^{2}) + \sum_{\begin{array}{c} x, y \in X (r) \\ x \neq y \end{array}} (p_{Δ} (x, y) - p_{Δ}^{2}) .

The first sum on the RHS is bounded by

p_{Δ} | X (r) | = E [| Y |] \overset{(9.3)}{≲} L^{- \frac{1}{2} ε} {(E [| Y |])}^{2} .

Hence, we need to estimate the double sum only.

In the double sum, we consider separately the terms with d_L(x, y) ⩽ 2r, the terms with 2r < d_L(x, y) < L^−ɛ, the terms with L^−ɛ ⩽ d_L(x, y) ⩽ L^ɛ, and the terms with d_L(x, y) ⩾ L^ɛ.

a. The terms with d_L(x, y) ⩽ 2r.

Taking into account that the number of such pairs is $≲ | X (r) |$ ⁠, we bound this sum by $≲ p_{Δ} | X (r) | = E [| Y |] \overset{(9.3)}{≲} L^{- \frac{1}{2} ε} {(E [| Y |])}^{2}$ ⁠.

b. The terms with 2r < d_L(x, y) < L^−ɛ.

By the second observation, we conclude that w.o.p., this case cannot occur, that is, the probability that there exists a pair of almost-singular points $x, y \in Cr (\frac{1}{3} α, β, \frac{1}{2} Δ)$ with 2r < d_L(x, y) < L^−ɛ is O(L^−C) with any positive C. Thus, in this range, p_Δ(x, y) = O(L^−C), while the total number of pairs x, y is bounded by $| X (r) |^{2} ≃ {(L / r)}^{4} ≪ {(L / β)}^{4} ≪ L^{10}$ ⁠, provided that ɛ in the definition of the parameters β and α is sufficiently small. That is, the sum is negligibly small.

c. The terms with L^−ɛ ⩽ d_L(x, y) ⩽ L^ɛ.

In this case, we estimate each summand in the double sum by p(x, y), which, by the short-distance estimate in Lemma 15, is

≲ \max (d_{L} {(x, y)}^{- Θ}, 1) p^{2} ≲ L^{Θ ε} p_{Δ}^{2} .

Then, the whole double sum is

≪ L^{Θ ε} p_{Δ}^{2} | X (r) | \cdot {(L^{ε} / r)}^{2} ≲ {(p_{Δ} | X (r) |)}^{2} \cdot L^{- 2 + (2 + Θ) ε} = L^{- 2 + (2 + Θ) ε} {(E [| Y |])}^{2},

which gives what we needed with a large margin.

d. The terms with d_L(x, y) ⩾ L^ɛ.

In this case,

p_{Δ} (x, y) - p_{Δ}^{2} ⩽ p (x, y) - p_{Δ}^{2} = (p (x, y) - p^{2}) + (p^{2} - p_{Δ}^{2}) .

By the long-distance estimate (8.2) of the two-point function p(x, y),

p (x, y) - p^{2} ≲ L^{- γ ε} p^{2} ≲ L^{- γ ε} p_{Δ}^{2},

while, by Lemma 14,

p^{2} - p_{Δ}^{2} ≲ {(\log Δ)}^{1 / 4} Δ^{- 1 / 2} p^{2} ≲ L^{- ε / 10} p_{Δ}^{2} .

Thus, $p_{Δ} (x, y) - p_{Δ}^{2} ≲ L^{- c ε} p_{Δ}^{2}$ ⁠, and the whole double sum is bounded by $L^{- c ε} p_{Δ}^{2} \cdot | X (r) |^{2} = L^{- c ε} {(E [| Y |])}^{2}$ ⁠.

This completes the proof of the estimate of Var[ |Y| ] and hence that of Lemma 17.□

X. ASYMPTOTIC INDEPENDENCE

Lemma 18

(asymptotic independence). Let (f_L) be a regular Gaussian ensemble, and let

Z \subset S^{2} (L)

be an L^1−κ-separated set with sufficiently small positive κ. Then, there exist a collection (ξ(z))_z∈Z of independent standard Gaussian random variables and positive constants c₁, c₂ so that, for L ⩾ L₀,

P \{\max_{z \in Z} | f_{L} (z) - ξ (z) | > L^{- c_{1}}\} < e^{- L^{c_{2}}} .

Proof.

We will follow rather closely the Proof of Theorem 3.1 in Ref. 5. We fix sufficiently large L and introduce the following notation:

$H_{f_{L}}$ is a Gaussian Hilbert space generated by f_L, i.e., the closure of finite linear combinations ∑c_jf_L(x_j) with the scalar product generated by the covariance.
$H$ is “a big Gaussian Hilbert space” that contains $H_{f_{L}}$ and countably many mutually orthogonal one-dimensional subspaces that are orthogonal to $H_{f_{L}}$ ⁠.
J(z) = f_L(z), z ∈ Z, are unit vectors in $H_{f_{L}}$ with

|〈 J (z), J {(z^{'}) 〉}_{H}| = |E [f_{L} (z) \bar{f_{L} (z^{'})}]| ≲ L^{- γ (1 - κ)}, z \neq z^{'} .

We claim that there exists a collection of orthonormal vectors

{\{\tilde{J} (z)\}}_{z \in Z} \subset H

such that

\max_{z \in Z} ‖ \tilde{J} (z) - J (z) ‖_{H} ⩽ 2 L^{- 0.45 γ} .

To prove this claim, we consider the Hermitian matrix Γ = Γ(z, z′)_z,z′∈Z with the elements

Γ (z, z^{'}) = \{\begin{cases} - {⟨ J (z), J (z^{'}) ⟩}_{H}, & z^{'} \neq z, \\ L^{- 0.9 γ}, & z^{'} = z . \end{cases})

For L > L₀(γ, κ), the matrix Γ is positive-definite. Indeed, by the classical Gershgorin theorem, each eigenvalue of Γ lies in one of the intervals (Γ(z, z) − t(z), Γ(z, z) + t(z)) with

t (z) = \sum_{z^{'} \in Z \ {z}} | Γ (z, z^{'}) |,

so we need to check that, for each z ∈ Z, t(z) < Γ(z, z), which is easy to see since

t (z) < | Z | \cdot L^{- γ (1 - κ)} \overset{| Z | ≲ L^{2 κ}}{≲} L^{2 κ} \cdot L^{- γ (1 - κ)} ≪ L^{- 0.9 γ},

provided that κ is sufficiently small. Since the Hermitian matrix Γ is positive-definite, we can find a collection of vectors

{I (z)}_{z \in Z} \subset H ⊖ span {J (z)}_{z \in Z}

with the Gram matrix Γ. Note that

‖ I (z) ‖_{H} = \sqrt{Γ (z, z)} = L^{- 0.45 γ}

⁠. Then, we let

\tilde{J} (z) = \frac{J (z) + I (z)}{‖ J (z) + I (z) ‖_{H}}, z \in Z .

By construction, the system of vectors

{\tilde{J} (z)}_{z \in Z}

is orthonormal in

H

⁠. Furthermore,

1 ⩽ ‖ J (z) + I (z) ‖_{H} ⩽ 1 + L^{- 0.45 γ},

whence

‖ J (z) - \tilde{J} (z) ‖_{H} ⩽ (‖ J (z) + I (z) ‖_{H} - 1) ‖ J (z) ‖_{H} + ‖ I (z) ‖_{H} ⩽ 2 L^{- 0.45 γ},

proving the claim.

It remains to note that, for each z ∈ Z, we have

P \{| \tilde{J} (z) - J (z) | > t\} = e^{- \frac{1}{2} {(t ‖ \tilde{J} (z) - J (z) ‖_{H}^{- 1})}^{2}} ⩽ e^{- \frac{1}{8} t^{2} L^{0.9 γ}},

whence, by the union bound,

P \{\max_{z \in Z} | \tilde{J} (z) - J (z) | > t\} ⩽ | Z | e^{- \frac{1}{8} t^{2} L^{0.9 γ}} ≲ L^{2 κ} e^{- \frac{1}{8} t^{2} L^{0.9 γ}} .

Letting t = L^−0.4γ, we complete the Proof of Lemma 18.□

XI. TWO SIMPLE LEMMAS

In this section, we present two simple and standard lemmas, which will be employed later. To keep this work relatively self-contained, we will include their proofs.

A. Anticoncentration of the sums of Bernoulli random variables

Lemma 19.

Given

0 < p_{0} ⩽ \frac{1}{2}

, let (η_j) be a collection of N independent random variables on the probability space Ω such that η_j attains the value 1 with probability p_j, p₀ ⩽ p_j ⩽ 1 − p₀, and the value 0 with probability 1 − p_j. Let

S_{N} = \sum_{j = 1}^{N} η_{j} .

Then, there exists ɛ = ɛ(p₀) > 0 such that for any measurable function, Q: Ω → [0, 1] with

\int_{Ω} Q d P ⩾ 1 - ε

, and for any

m \in R

, we have

\int_{Ω} | S_{N} - m |^{2} Q d P ⩾ c (p_{0}) N .

Proof of Lemma 19.

During the proof, the value of c(p₀) may vary from line to line. Take

λ = 1 / \sqrt{N}

⁠. First, we claim that

| E [e^{i λ S_{N}}] | ⩽ 1 - c (p_{0})

⁠. Indeed,

| E [e^{i λ η_{j}}] | = | e^{i λ} \cdot p_{j} + 1 \cdot (1 - p_{j}) | ⩽ 1 - c (p_{0}) λ^{2},

whence

|E [e^{i λ S_{N}}]| ⩽ {(1 - c (p_{0}) λ^{2})}^{N} ⩽ e^{- c (p_{0}) N λ^{2}} ⩽ 1 - c (p_{0}) .

Therefore, for any

m \in R

⁠,

|E [e^{i λ (S_{N} - m)} - 1]| ⩾ c (p_{0}) .

Next, using that |e^it − 1| ⩽ |t|, we proceed as follows:

\begin{array}{l} c (p_{0}) & ⩽ |E [(e^{i λ (S_{N} - m)} - 1) \cdot (Q + (1 - Q))]| \\ ⩽ \int_{Ω} λ | S_{N} - m | \cdot Q d P + \int_{Ω} 2 (1 - Q) d P \\ ⩽ \sqrt{\frac{1}{N} \int_{Ω} | S_{N} - m |^{2} \cdot Q d P} + 2 ε . \end{array}

Taking ɛ ⩽ c(p₀)/4, we complete the proof.□

B. Large sections lemma

Lemma 20.

Let

(Ω_{1} \times Ω_{2}, P_{1} \times P_{2})

be a product probability space, and let 0 < p ⩽ 1 and

0 < ε ⩽ \frac{1}{2} p

. Let Q: Ω₁ × Ω₂ → [0, 1] be a measurable function with

\int_{Ω_{1} \times Ω_{2}} Q d P ⩾ 1 - ε

, and let X ⊂ Ω₁ be an event with

P_{1} (X) ⩾ p

. Then,

P_{1} \{ω_{1} \in X : \int_{Ω_{2}} Q (ω_{1}, ω_{2}) d P_{2} (ω_{2}) ⩾ 1 - 2 ε p^{- 1}\} ⩾ \frac{p}{2} .

Proof of Lemma 20.

Put

X^{'} = \{ω_{1} \in X : \int_{Ω_{2}} Q (ω_{1}, ω_{2}) d P_{2} (ω_{2}) ⩾ 1 - 2 ε p^{- 1}\} .

Then,

\begin{array}{l} 1 - ε & ⩽ \int_{Ω_{1} \times Ω_{2}} Q d P \\ = (\int_{Ω_{1} \ X} + \int_{X^{'}} + \int_{X \ X^{'}}) (\int_{Ω_{2}} Q (ω_{1}, ω_{2}) d P_{2} (ω_{2})) d P_{1} (ω_{1}) \\ < 1 - P_{1} (X) + P_{1} (X^{'}) + (1 - 2 ε p^{- 1}) (P_{1} (X) - P_{1} (X^{'})) \\ ⩽ 1 - 2 ε + 2 ε p^{- 1} P_{1} (X^{'}), \end{array}

which yields the lemma.□

In the following, we will apply this lemma, mostly, with X = Ω₁ and p = 1.

XII. VARIANCE OF THE NUMBER OF LOOPS

Let G = G(V, E) be a finite graph embedded in the sphere $S^{2}$ with each vertex having degree four. We allow G to have multiple edges as well as “circular edges,” which connect a vertex with itself. The vertices of the graph are the joints (see Sec. VI D 2), the edges are curves on $S^{2}$ connecting the vertices, and the faces are the connected components of the open set $S^{2} \ (V \cup E)$ ⁠.

Each vertex v ∈ V can be replaced by one of two possible “avoided crossings” at v (Fig. 4).

FIG. 4.

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The vertex v and its states σ_v.

We call the choice of the avoided crossing at v the state of the vertex v and denote it by σ_v. When the states are assigned to all vertices in V, the collection of states σ_V = {σ_v: v ∈ V} turns the graphs G into a collection of loops Γ = Γ(σ_V).

We will deal with a random loop model when the states σ_v are independent random variables taking their values with probabilities p(v) and 1 − p(v). By $(Ω, P)$ ⁠, we denote the probability space on which the random states are defined. Then, N(Γ) = N(Γ(σ_V)) is a random variable on $(Ω, P)$ ⁠.

Given $0 < p_{0} ⩽ \frac{1}{2}$ ⁠, we put $V (p_{0}) = \{v \in V : p_{0} ⩽ p (v) ⩽ 1 - p_{0}\}$ and denote by |V(p₀)| the cardinality of the set of vertices V(p₀).

Lemma 21.

For any

0 < p_{0} ⩽ \frac{1}{2}

, there exist positive c(p₀), C(p₀), and ɛ = ɛ(p₀) such that for any function Q(σ_V) defined on the set of all possible states and taking the values in the interval [0, 1] with

\int_{Ω} Q (σ_{V}) d P ⩾ 1 - ε

and for any

m \in R

,

\int_{Ω} {(N (Γ (σ_{V})) - m)}^{2} Q (σ_{V}) d P ⩾ c (p_{0}) | V (p_{0}) |,

provided that |V(p₀)| ⩾ C(p₀).

A. Beginning the Proof of Lemma 21

We fix a function Q as above. In several steps, we will reduce the statement of the lemma to the anti-concentration bound for the sum of independent Bernoulli random variables provided by Lemma 19. In each of these steps, we will be using the following decoupling argument.

1. Decoupling

Suppose that the vertices are split into two disjoint parts V = V′⊔V^″ and decompose correspondingly $σ_{V} = (σ_{V^{'}}, σ_{V^{''}})$ ⁠. A collection of states σ_V′ assigned to the vertices from V′ generates (Fig. 5)

a collection Γ′ = Γ(σ_V′) of disjoint loops
and a graph G(σ_V′) with vertices at V^″, all of them having degree 4.

FIG. 5.

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Left: the graph G. Right: the loops Γ′ and the graph G(σ_V′).

By N(Γ′), we denote the number of loops in the collection Γ′. The collection of states $σ_{V^{''}}$ turns the graph G(σ_V′) into a collection of loops $Γ^{''} = Γ (σ_{V^{'}}, σ_{V^{''}})$ ⁠. Then, Γ = Γ′ ⊔Γ^″, and N(Γ) = N(Γ′) + N(Γ^″).

Let φ(σ_V) be any non-negative bounded measurable function. Since the random variables σ_V′ and $σ_{V^{''}}$ are independent, we have

\begin{array}{l} \int_{Ω} φ (σ_{V} (ω)) d P (ω) & = \iint_{Ω \times Ω} φ (σ_{V^{'}} (ω^{'}), σ_{V^{''}} (ω^{''})) d P (ω^{'}) d P (ω^{''}) . \\ = \int_{Ω} [\int_{Ω} φ (σ_{V^{'}} (ω^{'}), σ_{V^{''}} (ω^{''})) d P (ω^{''})] d P (ω^{'}) . \end{array}

Hence, for any event X′ ⊂Ω,

\int_{Ω} φ (σ_{V}) d P ⩾ P (X^{'}) \cdot \inf_{σ_{V^{'}} \in Σ^{'}} \int_{Ω} φ (σ_{V^{'}}, σ_{V^{''}} (ω^{''})) d P (ω^{''}),

where Σ′ = {σ_V′(ω′): ω′ ∈ X′}. Letting φ = (N(Γ) − m)²Q and taking into account that N(Γ) − m = N(Γ^″) − (m − N(Γ′)), we get

\begin{array}{l} \int_{Ω} {(N (Γ (σ_{V})) - m)}^{2} Q (σ_{V}) d P \\ ⩾ P (X^{'}) \cdot \inf_{σ_{V^{'}} \in Σ^{'}} \inf_{ℓ \in R} \int_{Ω} {(N (Γ^{''} (σ_{V^{''}} (ω^{''}))) - ℓ)}^{2} Q (σ_{V^{'}}, σ_{V^{''}} (ω^{''})) d P (ω^{''}) . \end{array}

The choice of the event X′ ⊂ Ω, or what is the same, of the set of states Σ′ is in our hands. Choosing it, we need to keep the value of the integral,

\inf_{σ_{V^{'}} \in Σ^{'}} \int_{Ω} Q (σ_{V^{'}}, σ_{V^{''}} (ω^{''})) d P (ω^{''}),

close to 1, while $P (X^{'})$ should stay bounded away from zero. This will be done with the help of Lemma 20. After that, it will suffice to prove Lemma 21 for the graph G(σ_V′) with vertices at V^″.

In the following, we will apply this decoupling argument several times. To simplify the notation, after each step, we treat the function Q as depending only on the states $σ_{V^{''}}$ of the remaining set of vertices V^″, ignoring its dependence on the fixed states σ_V′ ∈ Σ′.

B. Discarding the vertices v with p(v) < p₀ or p(v) > 1 − p₀

As above, we let $V (p_{0}) = \{v \in V : p_{0} ⩽ p (v) ⩽ 1 - p_{0}\}$ ⁠. We put V′ = V\V(p₀) and V^″ = V(p₀) and consider

X^{'} = \{ω^{'} \in Ω : \int_{Ω} Q (σ_{V^{'}} (ω^{'}), σ_{V^{''}} (ω^{''})) d P (ω^{''}) ⩾ 1 - 2 ε\} .

Then, by Lemma 20 (applied with p = 1), $P (X^{'}) ⩾ \frac{1}{2}$ ⁠.

Hence, from now on, we assume that for each vertex v of the graph G, we have p₀ ⩽ p(v) ⩽ 1 − p₀, while $\int_{Ω} Q (σ_{V}) d P ⩾ 1 - 2 ε$ ⁠.

C. Many faces have at most 4 vertices on the boundary

Denote by F the set of the faces of the graph G. Given a face $f \in F$ ⁠, we denote by $v (f)$ the set of vertices in V that lie on the boundary $\partial f$ ⁠. The Euler formula gives us

| V | - | E | + | F | = 1 + ν,

where ν is the number of connected components of the graph G. Since the degree of each vertex in G equals 4, we have |E| = 2|V|, whence |F| ⩾ |V| + 2. Furthermore, since each vertex in G lies on the boundary of at most four different faces, we have

| F | > | V | ⩾ \frac{1}{4} \sum_{f \in F} | v (f) | ⩾ \frac{1}{4} \sum_{\begin{array}{c} f \in F \\ | v (f) | ⩾ 5 \end{array}} | v (f) | ⩾ \frac{5}{4} |{f \in F : | v (f) | ⩾ 5}| .

We let $F^{*} = {f \in F : | v (f) | ⩽ 4}$ be the set of all faces having at most 4 vertices on the boundary and conclude that $|F^{*}| > \frac{1}{5} | F | > \frac{1}{5} | V |$ ⁠.

D. Choosing a maximal collection of separated faces

We call faces $f, f^{'} \in F^{*}$ separated if they do not have common vertices on their boundaries: $v (f) \cap v (f^{'}) = \emptyset$ ⁠. We fix a maximal collection $F \subset F^{*}$ of separated faces. Since for any face $f \in F$ ⁠, there are at most 12 other faces $f^{'} \in F^{*}$ with $v (f^{'}) \cap v (f) \neq \emptyset$ ⁠, we conclude from the maximality of $F$ that it also contains sufficiently many faces,

| F | ⩾ \frac{1}{13} |F^{*}| > \frac{1}{65} | V | .

E. Marking vertices and cycles

Next, we choose a simple cycle $c$ on the boundary of each face $f \in F$ and mark a vertex $v_{c}$ on that cycle according to the following rule: We take a vertex v′ in $v (f)$ and, starting at v′, walk along $\partial f$ turning left at each vertex so that the face $f$ always remains on the left-hand side. We stop when we return for the first time to the vertex that we already have passed and mark that vertex and the corresponding cycle (Fig. 6).

FIG. 6.

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Marking a cycle and a vertex on the boundary of the face $f$ ⁠.

We will be using the following property of the marked cycles: if the cycle exits in a vertex along an edge e, then it returns to this vertex along an edge, which is one of two edges adjacent to e (this follows from the fact that the same face cannot lie on both sides of some edge). This property yields that there exist states of vertices on $c$ ⁠, which turn $c$ into a separate loop.

We denote by V_M ⊂ V the set of all marked vertices and by V_UM = V\V_M the set of all unmarked vertices.

FIG. 7.

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Good and bad cycles.

FIG. 8.

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Separation vs merging.

F. Good marked cycles

For any particular assignment of the states of the unmarked vertices, the marked cycle $c$ is called good if the edges of $c$ merge into an edge of the graph G(σ(V_UM)) that connects the vertex $v_{c}$ with itself (Fig. 7). Note that this event depends only on the states of at most 3 unmarked vertices lying on $c \ {v_{c}}$ ⁠. Therefore, for any marked cycle $c$ ⁠, we have $P [c is good] ⩾ p_{0}^{3}$ ⁠. Hence, denoting by N_G the number of good cycles, we obtain

E [N_{G}] ⩾ p_{0}^{3} | {marked cycles} | = p_{0}^{3} | F | ⩾ \frac{1}{65} p_{0}^{3} | V | .

Using first the Chebyshev inequality and then the independence of the random states, we get

\begin{array}{l} P \{N_{G} ⩽ \frac{1}{2} E [N_{G}]\} & ⩽ \frac{4 Var [N_{G}]}{{(E [N_{G}])}^{2}} \\ ⩽ 13 0^{2} p_{0}^{- 6} \frac{| {marked cycles} |}{| V |^{2}} \\ ⩽ 13 0^{2} p_{0}^{- 6} | V |^{- 1} . \end{array}

To simplify the notation, we let $d (p_{0}) = \frac{1}{130} p_{0}^{3}$ and $D (p_{0}) = d {(p_{0})}^{- 2}$ ⁠. Then, letting $X = \{ω^{'} : N_{G} > d (p_{0}) | V |\}$ ⁠, we see that $P (X) ⩾ p \overset{def}{=} 1 - D (p_{0}) | V |^{- 1}$ ⁠. Put

X^{'} = \{ω^{'} \in X : \int_{Ω} Q (σ_{V_{UM}} (ω^{'}), σ_{V_{M}} (ω^{''})) d P (ω^{''}) ⩾ 1 - 4 ε p^{- 1}\}

(recall that at this moment $\int_{Ω} Q d P ⩾ 1 - 2 ε$ ⁠). Then, by Lemma 20, $P (X^{'}) ⩾ \frac{1}{2} p$ ⁠, which is $⩾ \frac{1}{4}$ ⁠, provided that |V| > 2D(p₀). From now on, we fix the states of the unmarked vertices corresponding to the event X′ and consider the remaining graph with vertices in V_M. At this step, $\int_{Ω} Q d P ⩾ 1 - 4 ε p^{- 1}$ ⁠.

G. Discarding bad cycles

All marked cycles $c$ are split into two classes: bad cycles and good cycles. Correspondingly, we decompose the set of all marked vertices V_M into the disjoint union V_M = V_M.B. ⊔ V_M.G. and consider

X^{'} = \{ω^{'} \in Ω : \int_{Ω} Q (σ_{V_{M . B .}} (ω^{'}), σ_{V_{M . G .}} (ω^{''})) d P (ω^{''}) ⩾ 1 - 8 ε p^{- 1}\}

By Lemma 20 (applied with X = Ω), $P (X^{'}) ⩾ \frac{1}{2}$ ⁠. We fix the states of marked bad vertices corresponding to the event X′.

H. Completing the Proof of Lemma 21

We are left with the graph G with vertices at V_M.G.. For each vertex v ∈ V_M.G., there is “a circular edge” e_v with the endpoints at v, which came from the corresponding cycle $c$ ⁠. Each state σ_v of the vertex v either creates from this circular edge a separate loop or merges it with other edges (Fig. 8).

We fix a collection of states $σ_{V_{M . G .}}^{*}$ of good marked vertices such that that none of the corresponding good cycles turns into a separate loop and denote by $Γ (σ_{V_{M . G .}}^{*})$ the loop ensemble obtained from the graph G after the assignment of the states $σ_{V_{M . G .}}^{*}$ ⁠. Introduce a collection of independent Bernoulli random variables ${\{η_{v}\}}_{v \in V_{M . G .}}$ ⁠, letting η_v = 0 if $σ_{v} = σ_{V_{M . G .}}^{*} (v)$ and η_v = 1 otherwise. Then,

N (Γ (σ_{V_{M . G .}})) = N (Γ (σ_{V_{M . G .}}^{*})) + \sum_{v \in V_{M . G .}} η_{v} .

Applying Lemma 19, we get the uniform in m lower bound,

\int_{Ω} {(N (Γ (σ_{V_{M . G .}})) - m)}^{2} Q (σ_{V_{M . G .}}) d P ⩾ c (p_{0}) | V_{M . G .} | .

To finish off the Proof of Lemma 21, it remains to recall that good marked vertices are in the one-to-one correspondence with good cycles, that is, |V_M.G.| = N_G, and that the states of unmarked vertices were fixed so that N_G ⩾ d(p₀)|V|.□

XIII. TYING LOOSE ENDS TOGETHER: PROOF OF THE THEOREM

A. Perturbing f_L

We choose a sufficiently small ɛ > 0 and take α′ = L^−2+ɛ and α = L^−2+2ɛ. Then, we take the function f_L and its independent copy g_L and put

{\tilde{f}}_{L} = \sqrt{1 - α^{' 2}} f_{L} + α^{'} g_{L} .

This is a random Gaussian function equidistributed with f_L. We will show that

\inf_{m \in R} E [{(N ({\tilde{f}}_{L}) - m)}^{2}] ≳ L^{c ε},

which immediately yields the lower bound for Var[N(f_L)] we are after. Note that

E [{(N ({\tilde{f}}_{L}) - m)}^{2}] = E^{f_{L}} E^{g_{L}} [N (\sqrt{1 - α^{' 2}} f_{L} + α^{'} g_{L}) - m {()}^{2}] .

That is, it suffices to show that with probability at least $\frac{1}{2}$ in f_L, we have

E^{g_{L}} [{(N ({\tilde{f}}_{L}) - m)}^{2}] ≳ L^{c ε} .

B. Freezing f_L

We will prove a somewhat stronger statement that this inequality holds if the function f_L satisfies the following conditions:

f_L ∈ C³(A, Δ, α, β) (introduced in Sec. VI A) with A = log L, Δ = L^3ɛ, and with β chosen so that β²L^7ɛ = α (i.e., $β = L^{- 1 - \frac{5}{2} ε}$ ⁠),
the set Cr(α) is L^1−Cɛ-separated, and
|Cr(α′)| ⩾ L^cɛ.

By Lemmas 13 and 17, these three conditions hold w.h.p. in f_L. From now on, we fix f_L so that these conditions hold and omit the index g_L, meaning $P = P^{g_{L}}$ ⁠, $E = E^{g_{L}}$ ⁠, etc.

C. Recalling a little Morse caricature

The rest will essentially follow from our little Morse caricature summarized in Lemma 12 combined with Lemma 21 on the fluctuations in the number of random loops. In order to apply Lemma 12, first, we observe that the relations

A α^{'} ≪ α, A Δ^{2} β^{2} ≪ α ≪ {(A Δ)}^{- 2} β, A^{2} Δ^{3} α ≪ 1,

required in Lemma 12 readily follow from our choice of the parameters α, α′, β, A, and Δ made few lines above. Lemma 12 also needs the lower bound

\min_{Cr (α)} | {\tilde{f}}_{L} | ≳ A Δ^{2} α^{2},

which holds w.h.p. with a large margin since AΔ²α² = L^−4+10ɛ log L, while, as we will momentarily see, w.h.p. in g_L, we have

\min_{Cr (α)} | {\tilde{f}}_{L} | > α^{'} L^{- c_{1}},

(13.1)

where c₁ ⩽ 1 is a constant from Lemma 18 (recall that α′ = L^−2+ɛ). Indeed, since g_L(p) is a standard Gaussian random variable, the probability that

| \sqrt{1 - α^{' 2}} f_{L} (p) + α^{'} g_{L} (p) | ⩽ α^{'} L^{- c_{1}}

at a given point p is $≲ L^{- c_{1}}$ ⁠. By the union bound, the probability that this happens somewhere on Cr(α) is

≲ L^{- c_{1}} | Cr (α) | ≲ L^{- c_{1} + 2 C ε} ≪ L^{- c_{1} / 2},

provided that ɛ is sufficiently small.

Thus, Lemma 12 applied to the functions f_L and ${\tilde{f}}_{L}$ yields that

N ({\tilde{f}}_{L}) = N_{I} ({\tilde{f}}_{L}) + N_{II} ({\tilde{f}}_{L}) + N_{III} ({\tilde{f}}_{L})

on the major part of the probability space where g_L ∈ C³(log L) and where estimate (13.1) holds. The first term on the RHS, $N_{I} ({\tilde{f}}_{L})$ ⁠, comes from the stable connected components of Z(f_L). Hence, on the large part of the probability space, the fluctuations in $N ({\tilde{f}}_{L})$ come only from the blinking circles $N_{II} ({\tilde{f}}_{L})$ and from the Bogomolny–Schmit loops $N_{III} ({\tilde{f}}_{L})$ ⁠, and after we have fixed the function f_L, both these quantities depend only on the configuration of (random) signs of ${\tilde{f}}_{L} (p)$ and p ∈ Cr(α).

D. Fight for independence

To make these random signs independent, using Lemma 18, we choose a collection of independent standard Gaussian random variables ξ(p), p ∈ Cr(α), so that

\max_{p \in Cr (α)} | g_{L} (p) - ξ (p) | ⩽ L^{- c_{1}} .

(13.2)

Denote by Ω′ the event that $‖ g_{L} ‖_{C^{3}} ⩽ \log L$ ⁠, and both estimates (13.1) and (13.2) hold. Then,

τ (L) \overset{def}{=} P (Ω \ Ω^{'}) = o (1), L \to \infty,

while on Ω′, we have

sgn ({\tilde{f}}_{L} (p)) = s (p) \overset{def}{=} sgn (\sqrt{1 - α^{' 2}} f_{L} (p) + α^{'} ξ (p)), p \in Cr (α) .

Note that the random signs s(p) are independent and that there exists $p_{0} \in (0, \frac{1}{2}]$ such that on Cr(α′) each of the two possible values of s(p) is attained with probability at least p₀.

For any subset Z ⊂ Cr(α), we let $s_{Z} = {(s (p))}_{p \in Z}$ ⁠. As before, we use the notation

\begin{array}{l} {Cr}_{S} (α) & = {p \in Cr (α) : p is a saddle point of f_{L}}, \\ {Cr}_{E} (α) & = {p \in Cr (α) : p is a local extremum of f_{L}} . \end{array}

Since, on Ω′, $N_{II} ({\tilde{f}}_{L})$ depends only on $s_{{Cr}_{E} (α)}$ and $N_{III} ({\tilde{f}}_{L})$ on $s_{{Cr}_{S} (α)}$ ⁠, there exist functions ${\tilde{N}}_{II} (s_{{Cr}_{E} (α)})$ and ${\tilde{N}}_{III} (s_{{Cr}_{S} (α)})$ such that, on Ω′, we have $N_{II} ({\tilde{f}}_{L}) = {\tilde{N}}_{II} (s_{{Cr}_{E} (α)})$ and $N_{III} ({\tilde{f}}_{L}) = {\tilde{N}}_{III} (s_{{Cr}_{S} (α)})$ ⁠. The function $N_{I} ({\tilde{f}}_{L})$ stays constant on Ω′, and by ${\tilde{N}}_{I}$ ⁠, we denote the value of that constant. Denoting by χ_Ω′ the indicator function of the event Ω′, we get

\begin{array}{l} E [{(N ({\tilde{f}}_{L}) - m)}^{2}] & ⩾ \int_{Ω} {[N_{II} ({\tilde{f}}_{L}) + N_{III} ({\tilde{f}}_{L}) - (m - N_{I} ({\tilde{f}}_{L}))]}^{2} χ_{Ω^{'}} d P \\ = \int_{Ω} {[{\tilde{N}}_{II} + {\tilde{N}}_{III} - (m - {\tilde{N}}_{I})]}^{2} E [χ_{Ω^{'}} |) s_{Cr (α)}] d P . \end{array}

The conditional expectation $E [χ_{Ω^{'}} |) s_{Cr (α)}]$ can be written as Q(s_Cr(α)), where Q is a function on a finite set S_Cr(α) of all possible collections of signs s_Cr(α). Thus,

E [{(N ({\tilde{f}}_{L}) - m)}^{2}] ⩾ \int_{Ω} {[{\tilde{N}}_{II} + {\tilde{N}}_{III} - (m - {\tilde{N}}_{I})]}^{2} Q d P .

Note that $E [Q] = P (Ω^{'}) = 1 - τ (L)$ ⁠.

E. Fluctuations generated by the Bogomolny–Schmit loops

First, we consider the case when $| {Cr}_{S} (α^{'}) | ⩾ \frac{1}{2} | Cr (α^{'}) |$ and look at the fluctuations in the number of the Bogomolny–Schmit loops. We use a decoupling argument similar to the one introduced in Sec. XII A 1. We decompose $s_{Cr (α)} = (s_{{Cr}_{E} (α)}, s_{{Cr}_{S} (α)})$ and let

X^{'} = \{ω_{1} \in Ω : \int_{Ω} Q (s_{{Cr}_{E} (α)} (ω_{1}), s_{{Cr}_{S} (α)} (ω_{2})) d P (ω_{2}) ⩾ 1 - 2 τ (L)\} .

Then, by Lemma 20, $P (X^{'}) ⩾ \frac{1}{2}$ ⁠, and therefore,

\begin{array}{l} E [{(N ({\tilde{f}}_{L}) - m)}^{2}] \\ ⩾ \frac{1}{2} \underset{ω_{1} \in X^{'}}{ess inf} \int_{Ω} {[{\tilde{N}}_{III} (s_{{Cr}_{S}} (ω_{2})) - (m - {\tilde{N}}_{I} - {\tilde{N}}_{II} (s_{{Cr}_{E}} (ω_{1})))]}^{2} Q (s_{{Cr}_{E}} (ω_{1}), s_{{Cr}_{S}} (ω_{2})) d P (ω_{2}) . \end{array}

We fix ω₁ ∈ X′ and the corresponding signs $s_{{Cr}_{E} (α)} (ω_{1})$ and consider the graph G(V, E) introduced in (6.6). The vertices of this graph are the joints J(p, δ) with p ∈ Cr_S(α) and δ = c(AΔ)⁻¹ with sufficiently small positive constant c. The edges are connected components of the set

Z (f_{L}) \ ⋃_{p \in {Cr}_{S} (α)} J (p, δ)

that touch the boundaries ∂J(p, δ). The random states σ_v are defined by the signs s(p), and, by construction, are independent. Furthermore, for the vertices v corresponding to the set Cr_S(α′), the probabilities of the states σ_v lie in the range [p₀, 1 − p₀]. Then, Lemma 21 yields that

\inf_{m \in R} E [{(N ({\tilde{f}}_{L}) - m)}^{2}] ≳ | {Cr}_{S} (α^{'}) | ⩾ \frac{1}{2} L^{c ε} .

This finishes the proof of our theorem in the case when $| {Cr}_{S} (α^{'}) | ⩾ \frac{1}{2} | Cr (α^{'}) |$ ⁠.

F. Fluctuations generated by blinking circles

It remains to consider the case when at least half of the critical points in Cr(α′) are local extrema. In this case, we use the decomposition Cr(α) = (Cr(α) \Cr_E(α′)) ⊔ Cr_E(α′) and once again combine Lemma 20 with Lemma 19. Let

X^{'} = \{ω_{1} \in Ω : \int_{Ω} Q (s_{Cr (α) \ {Cr}_{E} (α^{'})} (ω_{1}), s_{{Cr}_{E} (α^{'})} (ω_{2})) d P (ω_{2}) ⩾ 1 - 2 τ (L)\} .

Then, by Lemma 20, $P (X^{'}) ⩾ \frac{1}{2}$ ⁠. We fix ω₁ ∈ X′ and the corresponding $s_{Cr (α) \ {Cr}_{E} (α^{'})} (ω_{1})$ ⁠. The value of ${\tilde{N}}_{II}$ is the number of p ∈ Cr_E(α′) such that the sign s(p) is opposite to the sign of the eigenvalues of $H_{f_{L}} (p)$ ⁠. To each p ∈ Cr_E(α′), we associate a Bernoulli random variable

η_{p} = \{\begin{cases} 1, & s (p) is opposite to the sign of the eigenvalues of H_{f_{L}} (p), \\ 0, & otherwise . \end{cases})

Since the signs s(p) are independent, the variables η_p are independent as well. Recall that everywhere on Cr(α′), each of two possible values of s(p) is attained with probability ⩾p₀, and note that

{\tilde{N}}_{II} = \sum_{p \in {Cr}_{E} (α^{'})} η_{p} .

Then, Lemma 19 does the job. This finishes off the proof of our theorem in the second case when $| {Cr}_{E} (α^{'}) | ⩾ \frac{1}{2} | Cr (α^{'}) |$ ⁠.□

XIV. THE CASE OF SPHERICAL HARMONICS

As we have already mentioned in the Introduction, our theorem does not straightforwardly apply to the ensemble of Gaussian spherical harmonics. Here, we will outline minor modifications needed in this case.

The spherical harmonic f_n is an even (when its degree n is even) or odd (when n is odd) function. Hence, its zero set Z(f_n) is symmetric with respect to the origin. Hence, for the advanced readers, we are just working on the projective space $R P^{2}$ instead of the sphere. For the rest of the readers, the critical points of f_n come in symmetric pairs.

Instead of distances between critical points, we now have to talk about distances between symmetric pairs of points on $S^{2} (n)$ ⁠.

The values f_n(p) and f_n(−p) are equal up to the sign (+ if f_n is even and − if f_n is odd), while for n^1−Cɛ-separated pairs (p₁, −p₁) and (p₂, −p₂), the random variables f_n(p₁) and f_n(p₂) are almost independent.

When applying Lemma 1 to replace g_n(p) by ξ(p), we keep the relation between ξ(p) and ξ(−p) the same as between g_n(p) and g_n(−p), i.e., they coincide up to a sign, and make ξ(p) and ξ(p′) independent for p ≠ ± p′.

The rest of the argument goes as before with one simplification and two minor caveats.

The simplification is that for the spherical harmonics ensemble, the blinking circles cannot occur: by the classical Faber–Krahn inequality, the area of any nodal domain of a spherical harmonic of degree n on the sphere $S^{2} (n)$ cannot be less than a positive numerical constant, while the blinking circles are contained in the spherical disks D(p, δ), p ∈ Cr(α), of radius δ = c(AΔ)⁻¹ = o(1) as n → ∞. Another way to see that there are no blinking circles is to recall that all local minima of a spherical harmonic are negative, and all local maxima are positive. Let the signs of the eigenvalues of the Hessian $H_{f_{n}} (p)$ ⁠, p ∈ Cr(α), coincide, suppose that they are positive, that is, p is a local minimum of f_n, and therefore, f_n(p) < 0. By Lemma 6, f_n and ${\tilde{f}}_{n}$ remain positive on ∂D(p, δ) and convex in D(p, δ). By Lemma 5, ${\tilde{f}}_{n}$ has a critical point p_t ∈ D(p, δ), which is its local minimum. If the blinking circle occurs, when it disappears, ${\tilde{f}}_{n}$ should stay positive everywhere in D(p, δ), in particular, at p_t, yielding the contradiction. Hence, we need to treat only the Bogomolny–Schmit loops.

Both caveats pertain to the Proof of Lemma 21, which estimates from below the fluctuations in the number of random loops. First of all, we note that since all steps of our construction were symmetric with respect to the mapping x ↦ −x, the results it produces are also symmetric. In particular, the joints J(p, δ) and J(−p, δ) are symmetric, and the set of connected components of

Z (f_{L}) \ ⋃_{p \in {Cr}_{S} (α)} J (p, δ)

that touch the boundaries ∂J(p, δ) is also symmetric. Hence, the graph G(V, E), to which Lemma 21 was applied, is symmetric as well.

When defining marked cycles, we cannot choose a cycle ℓ passing through antipodal vertices, i.e., having common vertices with the symmetric cycle −ℓ. Fortunately, this does not happen often: there are at most eight faces $f$ such that $f$ and $- f$ have a common vertex v (the case in which −v will be also a common vertex of $f$ and $- f$ ⁠). This follows from the following lemma.

Lemma 22.

Let $X, X^{'} \subset S^{2}$ be two closed symmetric (with respect to the inversion x ↦ −x) non-empty symmetric sets. Then, X ∩ X′ ≠ ∅.

First, we conclude our argument and then will prove Lemma 22. Let $f$ and $- f$ have a pair of common vertices v and −v on their boundaries, and then, we can join v and −v by a path γ with $γ \ {v, - v} \subset f$ and by the path −γ with $- γ \ {v, - v} \subset - f$ ⁠. Put X = γ ∪ −γ. This is a symmetric closed connected subset of $S^{2}$ ⁠. If $f^{'}$ is another such face (different from $\pm f$ ⁠), then we have another symmetric closed connected set X′, and by Lemma 22, X ∩ X′ ≠ ∅. Since (γ ∪ −γ) \{v, −v} is contained inside $f \cup - f$ ⁠, while $(f \cup - f) \cap (f^{'} \cup - f^{'}) = \emptyset$ ⁠, we see that v and −v must be vertices on $\partial f^{'} \cup \partial (- f^{'})$ as well. Recalling that each vertex on our graph has degree 4, we see that there are at most eight such “bad faces” $f$ ⁠.

Proof of Lemma 22.

Assume that X₁ ∩ X₂ = ∅. Without loss of generality, we assume that the set X₂ contains the North and the South Poles. Since X₁ and X₂ are compact, there exists ɛ > 0 such that dist(X₁, X₂) > 6ɛ.

Take any point z ∈ X₁. Then, −z ∈ X₁ as well. Since X₁ is connected, there exists a finite chain of points in X₁ z = z₀, z₁, …, z_n = −z, with d(z_j, z_j−1) < ɛ (j = 1, …, n). Connecting z_j−1 to z_j by the shortest arc, we get a curve γ₁ joining z to −z and staying in the ɛ-neighborhood of X₁. Let γ = γ₁ ∪ (−γ₁). Then, γ is a symmetric curve going from z to −z and back and staying in the ɛ-neighborhood of X₁.

In a similar way, we can construct a curve from the North Pole to the South Pole staying in the ɛ-neighborhood of X₂. Let γ₂ be the piece of that curve from the last intersection with the circle of radius ɛ around the North Pole to the first intersection with the circle of radius ɛ around the South Pole. Note that γ₂ and both these circles stay in the ɛ-neighborhood of X₂ and, thereby, are disjoint with γ.

Now, take the projection

(x_{1}, x_{2}, x_{3}) \mapsto (\frac{x_{1}}{\sqrt{x_{1}^{2} + x_{2}^{2}}}, \frac{x_{2}}{\sqrt{x_{1}^{2} + x_{2}^{2}}}, x_{3})

of

S^{2} \cap {| x_{3} | ⩽ \cos ε}

onto the cylinder

C = {x_{1}^{2} + x_{2}^{2} = 1, | x_{3} | ⩽ \cos ε}

⁠. Note that it preserves the symmetry with respect to the origin, so we get two disjoint curves

\tilde{γ}

and

{\tilde{γ}}_{2}

on this cylinder such that

\tilde{γ} = \tilde{γ_{1}} \cup (- \tilde{γ_{1}})

⁠, where

\tilde{γ_{1}}

goes from some point

\tilde{z} \in C

to

- \tilde{z}

and stays at positive distance from the edge circles {x₃ = ±cos ɛ} of C, while

{\tilde{γ}}_{2}

joins those edge circles.

Consider the universal covering map $p : S \to C$ ⁠, where $S = {(t, s) \in R^{2} : | s | ⩽ \cos ε}$ is a horizontal infinite strip and $p ((t, s)) = (\cos 2 π t, \sin 2 π t, s)$ ⁠. Note that, for $ℓ \in Z$ ⁠, $p ((t + \frac{1}{2} + ℓ, - s)) = - p ((t, s))$ ⁠. By the path lifting lemma, ${\tilde{γ}}_{2}$ is lifted to some curve Γ₂ on S joining the top and the bottom boundary lines. The curve ${\tilde{γ}}_{1}$ is lifted to some curve Γ₁(τ), τ ∈ [0, 1], joining (t₀, s₀) with (t₀ + ξ, −s₀), where $ξ \in Z + \frac{1}{2}$ ⁠. Then, the curve $Γ_{1}^{*} (τ) = (t (τ) + ξ, - s (τ))$ extends Γ₁ and projects to $- {\tilde{γ}}_{1}$ ⁠. This extension process can now be repeated and done in both directions, so we get a curve Γ in S staying away from the boundary and such that the first coordinate of Γ goes from −∞ to +∞ (if ξ < 0, we reorient Γ). We still have Γ ∩Γ₂ = ∅, so the increment of arg(w − w₂), as w runs over Γ and w₂ ∈ Γ₂ stays fixed, should not depend on w₂. However, this increment is +π when w₂ is on the top boundary line of S and −π when w₂ is on the bottom line. This contradiction proves the lemma.□

The second caveat is caused by the fact that good cycles now come in symmetric pairs, and the cycles in each pair simultaneously either merge other cycles or remain separate. Hence, our Bernoulli random variables η_p are now valued in {0, 2} instead of {0, 1}.

XV. DEDICATION

In memory of Jean Bourgain.

ACKNOWLEDGMENTS

We are grateful to Dmitry Belyaev, Ron Peled, Evgenii Shustin, and Boris Tsirelson for helpful discussions and suggestions.

This work was partially supported by the U.S. NSF (Grant No. DMS-1900008, F.N.) and ERC Advanced Grant No. 692616 (M.S.)

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Author(s)

Fluctuations in the number of nodal domains

I. INTRODUCTION

II. THE SETUP AND THE MAIN RESULT

III. MAIN STEPS IN THE PROOF

IV. NOTATION

V. PRELIMINARIES

A. Local coordinates

B. Statistical properties of the gradient and the Hessian

1. Independence

2. Non-degeneracy

3. Power decay of correlations

C. A priori smoothness of fL

VI. SMOOTH FUNCTIONS WITH CONTROLLED TOPOLOGY OF THE ZERO SET

A. The sets Cr(α), Cr(α, β), and Cr(α, β, Δ) and the class C3(A, Δ, α, β)

B. Near any almost singular point there is a unique critical point of f

C. Near any point p ∈ Cr(α) there is a unique critical point of ft

D. Local matters

1. Local extrema

2. Saddle points

E. Global matters: The gradient flow

F. The upshot

1. Stable loops

2. Blinking circles

3. The Bogomolny–Schmit loops

4. The main lemma of Sec. VI

VII. LOWER BOUNDS FOR THE HESSIAN OF fL ON THE ALMOST SINGULAR SET

VIII. THE TWO-POINT FUNCTION OF THE SET Cr(α, β)

A. Estimates of the two-point function

B. Proof of the short-distance estimate (8.1)

1. Beginning the proof

2. The normal and the tangential derivatives

3. Case (i): |a′′|2+|b′′|2⩾12

4. Case (ii): |a′|2+|b′|2⩾12

C. Proof of Lemma 16

1. Beginning the Proof of Lemma 16

2. Proof of Claim (a)

3. Proof of Claim (b)

4. Completing the Proof of Lemma 16

D. Proof of the long-distance estimate (8.2)

IX. STRUCTURE OF THE SET Cr(α) WITH L−2+ɛ ⩽ α ⩽ L−2+2ɛ

A. w.h.p., the set Cr(α) is L1−Cɛ-separated

B. w.h.p., |Cr(α)| ⩾ Lcɛ

1. Estimating E[|Y|]

2. Estimating Var[|Y|]

a. The terms with dL(x, y) ⩽ 2r.

b. The terms with 2r < dL(x, y) < L−ɛ.

c. The terms with L−ɛ ⩽ dL(x, y) ⩽ Lɛ.

d. The terms with dL(x, y) ⩾ Lɛ.

X. ASYMPTOTIC INDEPENDENCE

XI. TWO SIMPLE LEMMAS

A. Anticoncentration of the sums of Bernoulli random variables

B. Large sections lemma

XII. VARIANCE OF THE NUMBER OF LOOPS

A. Beginning the Proof of Lemma 21

1. Decoupling

B. Discarding the vertices v with p(v) < p0 or p(v) > 1 − p0

C. Many faces have at most 4 vertices on the boundary

D. Choosing a maximal collection of separated faces

E. Marking vertices and cycles

F. Good marked cycles

G. Discarding bad cycles

H. Completing the Proof of Lemma 21

XIII. TYING LOOSE ENDS TOGETHER: PROOF OF THE THEOREM

A. Perturbing fL

B. Freezing fL

C. Recalling a little Morse caricature

D. Fight for independence

E. Fluctuations generated by the Bogomolny–Schmit loops

F. Fluctuations generated by blinking circles

XIV. THE CASE OF SPHERICAL HARMONICS

XV. DEDICATION

ACKNOWLEDGMENTS

REFERENCES

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C. A priori smoothness of f_L

A. The sets Cr(α), Cr(α, β), and Cr(α, β, Δ) and the class C³(A, Δ, α, β)

C. Near any point p ∈ Cr(α) there is a unique critical point of f_t

VII. LOWER BOUNDS FOR THE HESSIAN OF f_L ON THE ALMOST SINGULAR SET

3. Case (i): $| a^{''} |^{2} + | b^{''} |^{2} ⩾ \frac{1}{2}$

4. Case (ii): $| a^{'} |^{2} + | b^{'} |^{2} ⩾ \frac{1}{2}$

IX. STRUCTURE OF THE SET Cr(α) WITH L^−2+ɛ ⩽ α ⩽ L^−2+2ɛ

A. w.h.p., the set Cr(α) is L^1−Cɛ-separated

B. w.h.p., |Cr(α)| ⩾ L^cɛ

1. Estimating $E [| Y |]$

a. The terms with d_L(x, y) ⩽ 2r.

b. The terms with 2r < d_L(x, y) < L^−ɛ.

c. The terms with L^−ɛ ⩽ d_L(x, y) ⩽ L^ɛ.

d. The terms with d_L(x, y) ⩾ L^ɛ.

B. Discarding the vertices v with p(v) < p₀ or p(v) > 1 − p₀

A. Perturbing f_L

B. Freezing f_L