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

with a positive numerical constant *c* and that the random variable *N*(*f*_{n}) exponentially concentrates around its mean.^{4} A beautiful Bogomolny–Schmit heuristics^{2} suggests that, for any *ɛ* > 0 and *n* large enough,

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

with some *σ* > 0. The upper bound with $\sigma =215$ 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 $S2$ 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 $S2$.

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 $S2(L)={x\u2208R3:|x|=L}$ of large radius *L* and is normalized by $E[fL(x)2]=1$ for all $x\u2208S2(L)$. 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

where *d*_{L} is the spherical distance on $S2(L)$. We call such an ensemble (*f*_{L}) *regular* if the following two conditions hold:

$C3+$

*-smoothness*: $KL\u2208C3+\nu ,3+\nu (S2(L))$ with estimates uniform in*L*and with some*ν*> 0.*Power decay of correlations*: $KL(x,y)\u2272(1+dL(x,y))\u2212\gamma $, $x,y\u2208S2(L)$, with some*γ*> 0 and with the implicit constant independent of*L*.

Condition (1) yields that almost surely, $fL\u2208C3(S2(L))$ 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 $S2(L)$. By *N*(*f*_{L}), we denote the number of these loops.

*Let*(

*f*

_{L})

*be a regular Gaussian ensemble. Then, there exists*

*σ*> 0

*such that, for*

*L*⩾

*L*

_{0}

*,*

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 $RP2$ 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

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

Then, as we will see, with high probability, given *f*_{L}, the change in the topology of the zero set $Z(f\u0303L)$ is determined by the signs of $f\u0303L$ at Cr(*α*), that is, by the collection of the random values $gL(p):p\u2208Cr(\alpha )$. To make the correlations between these random values negligible, the set Cr(*α*) should be well-separated on the sphere $S2(L)$. At the same time, the set Cr(*α*) has to be relatively large; otherwise, the impact of fluctuations in signs of $f\u0303L$ on the number $N(f\u0303L)$ will be negligible. In Lemma 17, we will show that

*there exist positive**ɛ*_{0}*,**c**, and**C**such that, given*0 <*ɛ*⩽*ɛ*_{0}*and**L*⩾*L*_{0}(*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 $f\u0303L$ at Cr(*α*) affect the topology of the zero set $Z(f\u0303L)$, 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 ∇^{2}*f*_{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 $Z(f\u0303L)$ depends only on the signs of the eigenvalues of the Hessian ∇^{2}*f*_{L}(*p*) and the signs of $f\u0303L(p)$, *p* ∈ Cr(*α*). We will describe how these signs determine the structure of the zero set $Z(f\u0303L)$ in small neighborhoods of the points *p* ∈ Cr(*α*). Outside these neighborhoods, the zero lines of $Z(f\u0303L)$ 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 ∇^{2}*f*_{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, $Z(f\u0303L)\u2229D(p,\delta )$ either consists of a simple loop encircling the point *p* when the sign of $f\u0303L(p)$ is opposite to that of the eigenvalues of ∇^{2}*f*_{L}(*p*) or is empty when these signs coincide. We call such connected components of $Z(f\u0303L)$ *blinking circles*.

Now, we turn to the case when the eigenvalues of the Hessian ∇^{2}*f*_{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 $S2(L)$. 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 ${sgn(f\u0303L(p)):p\u2208Cr(\alpha )}$ determines how the graph *G*(*f*_{L}) is turned into a collection of loops in $Z(f\u0303L)$, which we will call *the Bogomolny–Schmit loops*. Figure 1 illustrates how the sign of $f\u0303L$ at the saddle point *p* ∈ Cr(*α*) determines the structure of the zero set $Z(f\u0303L)$ in a small neighborhood of the saddle point *p* ∈ Cr(*α*).

We see that in both cases, *the fluctuations in the number of connected components of* $Z(f\u0303L)$ *are caused by fluctuations in the signs*,

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 $f\u0303L$ 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.$S2(L)$ denotes the sphere in $R3$ centered at the origin and of radius

*L*, while, as usual, $S2$ denotes the unit sphere in $R3$. By*d*_{L}, we denote the spherical distance on $S2(L)$. By $D(x,\rho )\u2282S2(L)$, we denote the open spherical disk of radius*ρ*centered at*x*.$G\xaf$ 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}($C3+\nu $-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 $p\u2208S2(L)$ its own coordinate chart. For $p\u2208S2(L)$, let Π_{p} be a plane in $R3$ passing through the origin and orthogonal to *p*. The Euclidean structure on Π_{p} is inherited from $R3$. By $Sp2(L)$, we denote the hemisphere of $S2(L)$ centered at *p*. By

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

whenever $|X|,|Y|\u2a7d12L$.

Let $f:S2(L)\u2192R$ be a smooth function. We put *F*_{p} = *f*◦Ψ_{p} and identify d^{k}*f*(*p*) with d^{k}*F*_{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 d*f*(*p*) (*v*) = ⟨∇*f*(*p*), *v*⟩, *v* ∈ Π_{p}, and the Hessian *H*_{f}(*p*) = ∇^{2}*f*(*p*) = ∇^{2}*F*_{p}(0) is a self-adjoint operator on Π_{p} such that d^{2}*f*(*p*) (*u*, *v*) = ⟨*H*_{f}(*p*)*u*, *v*⟩, *u*, *v* ∈ Π_{p}.

The notation $f\u2208Ck(S2(L))$ means that, for every $p\u2208S2(L)$, $f\u25e6\Psi p\u2208Ck{X\u2208\Pi p:|X|\u2a7d12L}$, and

Obviously, $\Vert f\Vert Ck\u2a7dmaxp\u2208S2(L)\Vert f\u25e6\Psi p\Vert Ck|X|\u2a7d12L$. In the other direction, it is not difficult to see that if $f\u2208Ck(S2(L))$, then $\Vert f\u25e6\Psi p\Vert Ck|X|\u2a7d12L\u2a7dCk\Vert f\Vert Ck$.

### B. Statistical properties of the gradient and the Hessian

Fix $p\u2208S2(L)$ and the orthogonal coordinate system (*X*_{1}, *X*_{2}) on the plane Π_{p}, and set $\u2202i1\u2026ikkf(p)=\u2202Xi1\u2026XikkFp(0)$, 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.

*Let* *f* *be a* *C*^{2+ν}*-smooth random Gaussian function on the sphere* $S2$ *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*∇^{2}*f*(*p*),*∂*_{1}*f*(*p*)*and**∂*_{2}*f*(*p*),*f*(*p*)*and*$\u22021,22f(p)$,*and*$\u22021,12f(p)$

*and*$\u22021,22f(p)$*, as well as*$\u22022,22f(p)$*and*$\u22021,22f(p)$*.*

*p*is the North Pole of the sphere $S2$ and suppress the dependence on

*p*, letting Ψ = Ψ

_{p}. Then,

*F*=

*f*◦Ψ is a Gaussian function on the unit disk {|

*X*

_{1}|

^{2}+ |

*X*

_{2}|

^{2}< 1} with the covariance

*F*(0), $\u2202XjF(0)$, and $\u2202XiXj2F(0)$. The covariances of these random variables can be computed using the relations

#### 2. Non-degeneracy

*Let*(

*f*

_{L})

*be a regular Gaussian ensemble, and let*$p\u2208S2(L)$

*. Then,*

*and*

Again, we assume that *p* is the North Pole of the sphere $S2(L)$. Put $\Psi L(X1,X2)=X1,X2,L2\u2212(X12+X22)$, and consider the Gaussian functions *F*_{L} = *f*_{L}◦Ψ_{L} defined in the disks $12LD$. The corresponding covariances $E[FL(X)FL(Y)]=KL(X,Y)$ are *C*^{3+ν,3+ν}-smooth on $12LD\xd712LD$ with some *ν* > 0. Their partial derivatives up to the third order are bounded locally uniformly in *L* ⩾ *L*_{0}. Hence, by a version of the Arzelá–Ascoli theorem, any sequence $KLj$ contains a locally uniformly *C*^{2+ν,2+ν}-convergent subsequence.

*K*is a

*C*

^{2+ν,2+ν}-smooth Hermitian-positive function on $R2\xd7R2$, 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*,

*L*⩾

*L*

_{0}, the limiting function

*K*satisfies

*ρ*cannot degenerate to the point measure at the origin.

*L*

_{j}→

*∞*,

*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

and

[we keep fixed the coordinate systems (*X*_{1}, *X*_{2}) and (*Y*_{1}, *Y*_{2}) in the planes Π_{p} and Π_{q}].

*Let*(

*f*

_{L})

*be a regular Gaussian ensemble. Then, for any*$p,q\u2208S2(L)$

*,*

Put $Kp,q(X,Y)=EFp(X)Fq(Y)$, where *F*_{p} = *f*_{L}◦Ψ_{p} and $(X,Y)\u2208Q\xaf\xd7Q\xaf$, where $Q\xaf=[\u22121,1]\xd7[\u22121,1]$. We have $E[Fp(0)\u2202YiFq(0)]=\u2202YiKp,q(0,0)$, and $E[\u2202XiFp(0)\u2202YjFq(0)]=\u2202XiYj2Kp,q(0,0)$.

There is nothing to prove if *d*_{L}(*p*, *q*) ⩽ 1, so we assume that *d*_{L}(*p*, *q*) ⩾ 1. Then, $\Vert Kp,q\Vert C(Q\xaf\xd7Q\xaf)\u2272d\u2212\gamma $ and $\Vert Kp,q\Vert C2,2(Q\xaf\xd7Q\xaf)\u22721$. Now, we use the classical Landau–Hadamard inequality in the following form: if $h:[0,1]\u2192R$ is a *C*^{2}-smooth function and *M*_{j} = max_{[0,1]}|*h*^{(j)}|, 0 ⩽ *j* ⩽ 2, then $M1\u2272max(M0,M0M2)$. Applying this to the functions *X*_{i}↦*K*_{p,q}(*X*, *Y*) and *Y*_{j}↦*K*_{p,q}(*X*, *Y*), we get $\Vert \u2202XiKp,q\Vert C(Q\xaf\xd7Q\xaf)\u2272d\u2212\gamma /2$ and $\Vert \u2202YjKp,q\Vert C(Q\xaf\xd7Q\xaf)\u2272d\u2212\gamma /2$, *i*, *j* = 1, 2. Applying the Landau–Hadamard inequality again, this time to the functions $Yj\u21a6\u2202XiKp,q(X,Y)$, we get $\Vert \u2202XiYj2Kp,q\Vert C(Q\xaf\xd7Q\xaf)\u2272d\u2212\gamma /4$, *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+dL(p,q))\u2212\gamma $.

### 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,*$\Vert fL\Vert C3(S2(L))<log\u2061L$.

This bound immediately follows from the classical estimate

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 *C*^{3}(*A*, Δ, *α*, *β*) of smooth functions *f* on $S2(L)$ such that the number of connected components of the zero set of a small perturbation $f\u0303$ of *f* can be recovered from the values of $f\u0303$ at the critical points of *f* with small critical values (provided that the values of $f\u0303$ 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 *C*^{3}**(***A*, Δ, *α*, *β***)**

Given *α*, *β* ⩽ 1 and Δ ⩾ 1, we let

and

where ‖⋅‖_{6op} stands for the operator norm.

By *C*^{3}(*A*), we denote the class of *C*^{3}-smooth functions *f* on $S2(L)$ with $\Vert f\Vert C3\u2a7dA$. Given the parameters

by *C*^{3}(*A*, Δ, *α*, *β*), we denote the class of functions *f* ∈ *C*^{3}(*A*) for which Cr(*α*, *β*) = Cr(*α*, *β*, Δ), i.e., the Hessian of *f* does not degenerate [$\Vert (\u22072f)\u22121\Vert op\u2a7d\Delta $] on the almost singular set Cr(*α*, *β*) where *f* and ∇*f* are simultaneously small.

Given *f* ∈ *C*^{3}(*A*, Δ, *α*, *β*), *α*′ ≪ *α*, and *g* ∈ *C*^{3}(*A*), we set

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*

*Suppose that*

*f*∈

*C*

^{3}(

*A*)

*and*

*p*∈ Cr(

*α*,

*β*, Δ)

*with*

*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*$Dp,c(A\Delta )\u22121$*, and*|

*f*(*z*)| ⩽ 2*α*,*provided that**A*Δ^{2}*β*^{2}≪*α*.

_{p}, and let

*F*=

*f*◦Ψ

_{p}and $HF=\u22072F$. To find the critical point

*z*, we use a simplified Newton method,

*X*|, |

*Y*| ⩽

*c*(

*A*Δ)

^{−1}with sufficiently small positive

*c*.

*D*(0,

*c*(

*A*Δ)

^{−1}). We have

*βA*Δ

^{2}≪ 1 and that the positive constant

*c*is sufficiently small.

*Z*in the disk

*D*(0,

*c*(

*A*Δ)

^{−1}), and

*Aβ*

^{2}Δ

^{2}≪

*α*.□

### C. Near any point *p* **∈** Cr**(***α***)** there is a unique critical point of *f*_{t}

*Let*

*f*,

*g*∈

*C*

^{3}(

*A*)

*, where*

*f*

_{t}=

*f*+

*tg*

*with*0 ⩽

*t*⩽

*α*′

*. Suppose that*

*p*∈ Cr(

*α*)

*with*$\Vert Hf(p)\u22121\Vert op\u2a7d\Delta $

*and that*

*Then, there exists a unique critical point*

*p*

_{t}

*of*

*f*

_{t}

*such that*

*d*

_{L}(

*p*,

*p*

_{t}) ≪ Δ

*α*

*and*|

*f*

_{t}(

*p*) −

*f*

_{t}(

*p*

_{t})| ≪

*A*(Δ

*α*)

^{2}

*. Moreover, there are no other critical points of*

*f*

_{t}

*at distance*⩽

*c*(

*A*Δ)

^{−1}

*from*

*p*

*.*

*f*

_{t}∈

*C*

^{3}(2

*A*) and that |

*f*

_{t}(

*p*)| ⩽ |

*f*(

*p*)| +

*Aα*′ and |∇

*f*

_{t}(

*p*)|≲

*Aα*′ ≪

*α*. Furthermore, $\Vert Hft(p)\u22121\Vert op\u2a7d2\Delta $. Indeed, we have

*f*

_{t}with

*β*=

*Aα*′), there exists a unique critical point

*p*

_{t}of

*f*

_{t}with

*d*

_{L}(

*p*,

*x*) ≪ Δ

*α*, we have

*d*

_{L}(

*p*

_{t},

*x*) ≪ Δ

*α*and then |∇

*f*

_{t}(

*x*)| = |∇

*f*

_{t}(

*x*) − ∇

*f*

_{t}(

*p*

_{t})| ≪

*A*⋅ Δ

*α*, whence

At last, by part B of Lemma 4, there are no other critical points of *f*_{t} at distance ⩽*c*(*A*Δ)^{−1} from *p*.□

### D. Local matters

Given *f* ∈ *C*^{3}(*A*), *p* ∈ Cr(*α*), and $\Vert Hf(p)\u22121\Vert op\u2a7d\Delta $, we look at the behavior of *f*_{t} in the *δ*-neighborhood of *p*. As above, *f*_{t} = *f* + *tg*, with *g* ∈ *C*^{3}(*A*) and 0 ⩽ *t* ⩽ *α*′. Throughout this section, we assume that the parameters *α*′, *α*, *δ*, *A*, and Δ satisfy the set of conditions

and that

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 $Hf(p)$ 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 $\Psi p\u22121D(p,\delta )\u2282\Pi p$.

*Suppose that the eigenvalues of the Hessian* $Hf(p)$ *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*$ft\u2202D(p,\delta )$*coincides with the sign of the eigenvalues of*$Hf(p)$*.*

*F*=

*f*◦Ψ

_{p}and

*F*

_{t}=

*f*

_{t}◦Ψ

_{p}, and suppose that $HF(0)=Hf(p)$ is positive definite (otherwise, replace

*f*by −

*f*), that is, $\u3008HF(0)x,x\u3009\u2a7e\Delta \u22121|x|2$. Then, for any $X\u2208\Psi p\u22121D(p,\delta )$, we have

*X*| ⩽

*δ*=

*c*(

*A*Δ)

^{−1}with sufficiently small

*c*. Noting that $\Vert HFt(X)\u2212HF(X)\Vert op\u2272A\alpha \u2032\u226a\Delta \u22121$, we get $\u3008HFt(X)x,x\u3009\u2a7e(4\Delta )\u22121|x|2$, which proves (i).

*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

*c*in (6.2) (the definition of

*δ*) was chosen so small that $C\u22c5c\u2a7d164$.□

Then,

*Z*(*f*_{t}) ∩*∂D*(*p*,*δ*) =*∅*.*Z*(*f*_{t}) ∩*D*(*p*,*δ*) is either empty, homeomorphic to $S1$, or a singleton.Suppose that |

*f*_{t}(*p*)| ≳*A*(Δ*α*)^{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 $Hf(p)$ have the same sign, and*Z*(*f*_{t}) ∩*D*(*p*,*δ*) is homeomorphic to $S1$ whenever*f*_{t}(*p*) and the eigenvalues of $Hf(p)$ 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 $Hf(p)$ 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 $H(X)=\u27e8HF(0)X,X\u27e9$, we denote the quadratic form generated by the Hessian $HF(0)$. WLOG, we assume that

We take *δ* = *c*(*A*Δ)^{−1} with a sufficiently small positive constant *c*, set

and call this set *a joint*. By

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

*J*(

*δ*), we have

*∂**

*J*(

*δ*), we have |

*H*| =

*aδ*

^{2}⩾

*c*

^{2}

*A*

^{−2}Δ

^{−3}, while

*Aδ*

^{3}=

*c*

^{3}

*A*

^{−2}Δ

^{−3}. This proves the lemma.□

The set $|H|\u2a7da\delta 2\u22c22\delta \u2a7d|X1|\u2a7d3\delta $ consists of four disjoint curvilinear quadrangles. We call them *terminals* and denote them by *T*_{i}, 1 ⩽ *i* ⩽ 4.

*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}*.*

*J*(

*δ*), we have

*T*

_{i},

*b*⩾

*a*⩾ Δ

^{−1}), and

*c*in the definition of

*δ*is sufficiently small. Furthermore,

*Aα*′ is also much smaller than Δ

^{−1}

*δ*(since

*Aα*′ ≪

*A*

^{−1}Δ

^{−2}). Thus, $|(Ft)X2|>0$ 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 $Xt=(X1t,X2t)$ of *F*_{t} and |*X*^{t}| ≪ Δ*α*. Consider the sets

Since

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

*Under the same assumptions as in Lemmas* 7 *and* 8*, the only extremum of the restriction of the function* *F*_{t} *to the segment* *I*_{1} *is a local minimum at* $X1=X1t$*, and the only extremum of the restriction of the function* *F*_{t} *to the segment* *I*_{2} *is a local maximum at* $X2=X2t$*.*

*A*Δ

^{2}

*α*≪ 1, and

*Aα*′ ≪

*α*), and the

*C*

^{3}-norm of

*F*

_{t}is bounded by

*CA*. Therefore, for $X1>X1t$, we have

*c*. Similarly, $(Ft)X1(X1,X2t)<0$ for $X1<X1t$.

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

*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})*.*

Since zero is not a critical value of the restriction $FtJ(\delta )$, 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 $G\u0304\u2282J(\delta )$. 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 Ω_{0}, 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 Ω_{0} 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_{p}*J*(*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_{p}*T*_{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*(Δ*α*)^{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* ∈ *C*^{3}(*A*, Δ, *α*, *β*) and *g* ∈ *C*^{3}(*A*). Let *f*_{t} = *f* + *tg* and $f\u0303=f\alpha \u2032$, and consider the gradient flow *z*_{t}, 0 ⩽ *t* ⩽ *α*′, defined by the ordinary differential equation (ODE),

with the initial condition *z*_{0} ∈ *Z*(*f*).

*Suppose that* *A*Δ^{2}*β*^{2} ≪ *α* ≪ (*A*Δ)^{−2}*β* *and* *Aα*′ ≪ *α**. Let* *δ* = *c*(*A*Δ)^{−1} *with a sufficiently small constant* *c* > 0*. Then, for any arc* *I* ⊂ *Z*(*f*) \⋃_{p∈Cr(α)}*D*(*p*, 2*δ*^{2})*, the flow* *z*_{t} *provides a* *C*^{1}*-homotopy of* *I* *onto an arc* $I\u0303\u2282Z(f\u0303)\\u22c3p\u2208Cr(\alpha )D(p,\delta 2)$*.* *Vice versa, for any arc* $I\u0303\u2282Z(f\u0303)\\u22c3p\u2208Cr(\alpha )D(p,2\delta 2)$*, the inverse flow* *z*_{α′−t} *provides a* *C*^{1}*-homotopy of* $I\u0303$ *onto an arc* *I* ⊂ *Z*(*f*) \⋃_{p∈Cr(α)}*D*(*p*, *δ*^{2})*. Moreover, these homotopies move the points by at most* *O*(*Aα*′/*β*)*.*

*ɛ*and

*η*has no importance), and let Ω be the interior of $\Omega \u0304$. Note that

*C*

^{1}-function on $\Omega \u0304$. Therefore, for any initial point $z0\u2208Z(f)\X\u0304$, the ODE has a unique

*C*

^{1}-solution. The solution exists until it reaches the boundary of Ω. Note that along the trajectory

*z*

_{t}, we have

*f*

_{t}(

*z*

_{t}) = 0 [recall that

*z*

_{0}∈

*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 $z\u0304$ with $|\u2207f(z\u0304)|\u2a7d\beta $. Recalling that the point

*z*

_{t}moves with the speed ⩽ 2

*A*/

*β*, we see that $dL(z\u0304,z0)\u2a7d2A\alpha \u2032/\beta $. Furthermore, by the continuity of

*f*

_{t}, we have $f\tau (z\u0304)=0$, whence $|f(z\u0304)|\u2a7d\alpha \u2032|g(z\u0304)|\u2a7dA\alpha \u2032\u226a\alpha $. Combining this with the gradient estimate $|\u2207f(z\u0304)|\u2a7d\beta $ and applying Lemma 4, we conclude that there is a unique critical point

*p*of

*f*with $dL(p,z\u0304)\u2a7d2\beta \Delta $, i.e., with

*δ*

^{2}.

*p*belongs to Cr(

*α*), which is straightforward,

*Aα*′ and

*A*Δ

^{2}

*β*

^{2}are both much less than

*α*.

Since the function $f\u0303=f+\alpha \u2032g$ belongs to the class *C*^{3}(2*A*, 2Δ, 2*α*, 2*β*), the same arguments can be also applied to the inverse flow *z*_{α′−t}.□

### F. The upshot

We start with functions *f* ∈ *C*^{3}(*A*, Δ, *α*, *β*) and *g* ∈ *C*^{3}(*A*) and consider the perturbation $f\u0303=f+\alpha \u2032g$. We assume that the parameters

satisfy the following relations:

[which particularly yield conditions (6.1)]. We also assume that the perturbation $f\u0303$ is not too small on Cr(*α*),

We set

We put *δ* = *c*(*A*Δ)^{−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 *c*_{0}(*A*Δ)^{−1}-separated with a positive constant *c*_{0}].

#### 1. Stable loops

These are connected components of *Z*(*f*) and $Z(f\u0303)$ that do not intersect the set

We denote by *N*_{I}(*f*) the number of stable loops in *Z*(*f*) and by $NI(f\u0303)$ the number of stable loops in $Z(f\u0303)$.

Observe that $Dp,ab\delta \u2282J(p,\delta )$, *p* ∈ Cr_{S}(*α*), and that $ab\u2a7e(A\Delta )\u22121/2$. We see that, for each *p* ∈ Cr(*α*), we have *D*(*p*, 2*δ*^{2}) ⊂ *U*. Therefore, Lemma 11 applies to stable loops in *Z*(*f*) as well as to stable loops in $Z(f\u0303)$ and yields a one-to-one correspondence between the set of stable loops in *Z*(*f*) and the set of stable loops in $Z(f\u0303)$. That is, $NI(f\u0303)=NI(f)$.

#### 2. Blinking circles

These are small connected components of $f\u0303$ 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, $Z(f\u0303)$ 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

#### 3. The Bogomolny–Schmit loops

This is the most interesting part of $Z(f\u0303)$. Consider the graph *G* = *G*(*f*) embedded in $S2(L)$. The vertices of *G* are the joints *J*(*p*, *δ*),*p* ∈ Cr_{S}(*α*). The edges are connected components of the set

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 $f\u0303(p)$, *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 $NIII(f\u0303)$, we denote the number of loops in this collection.

#### 4. The main lemma of Sec. VI

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

*Let*

*f*∈

*C*

^{3}(

*A*, Δ,

*α*,

*β*)

*,*

*g*∈

*C*

^{3}(

*A*),

*and*$f\u0303=f+\alpha \u2032g$

*. Suppose that the parameters*

*satisfy following relations*

*and that*

*Then,*

## VII. LOWER BOUNDS FOR THE HESSIAN OF *f*_{L} ON THE ALMOST SINGULAR SET

Now, we return to regular Gaussian ensembles (*f*_{L}).

*Given a sufficiently small positive*

*ɛ*

*, let*

*α*⩽

*L*

^{−2+2ɛ}

*and*

*β*

^{2}

*L*

^{3ɛ}⩽

*α*

*. Then, there exists*

*L*

_{0}=

*L*

_{0}(

*ɛ*)

*such that, for each*

*L*⩾

*L*

_{0}

*, w.h.p.,*

*where*‖⋅‖

_{op}

*denotes the operator norm.*

*ɛ*> 0 and consider the set Cr(5

*α*, 4

*β*). Let

*p*be the probability that a given point $x\u2208S2(L)$ belongs to the set Cr(5

*α*, 4

*β*). 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

Denote by *μ* = *μ*(*x*) the eigenvalue of the Hessian matrix ∇^{2}*f*_{L} with the minimal absolute value and by *w* = *w*(*x*) the corresponding normalized eigenvector. Assume that $\Vert fL\Vert C3<log\u2061L$ (recall that this holds w.o.p.), and suppose that, for some *x* ∈ Cr(*α*, *β*) and *L* ⩾ *L*_{0}, |*μ*(*x*)| < Δ^{−1}, where Δ = *L*^{3ɛ}. We will show that then, the set Cr(5*α*, 4*β*) contains a subset $G\u0303=G\u0303(x)$ with $area(G\u0303)\u2273\beta 2L3\epsilon (log\u2061L)\u22121$. This will immediately imply the lemma.

*x*∈ Cr(

*α*,

*β*) with |

*μ*(

*x*)| < Δ

^{−1}, take the corresponding map Ψ

_{x}, and let

*F*=

*f*

_{L}◦Ψ

_{x}. Put

*τ*=

*β*Δ ≪ 1. For

*Y*=

*tw*(

*x*), 0 ⩽

*t*⩽

*τ*, we have

*I*= [0,

*tw*(

*x*)], we get

*β*

^{2}Δ ⩽

*α*.

*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,

The next lemma gives us a lower bound for the probability that a given point $x\u2208S2(L)$ belongs to the set

Let *p* be the probability that a given point $x\u2208S2(L)$ belongs to the set Cr(*α*, *β*). By the invariance of the ensemble (*f*_{L}), this probability does not depend on *x*.

*For any*

*α*,

*β*⩽ 1

*and any*Δ ⩾ 2

*,*

*x*∈ Cr(

*α*,

*β*), the probability that

*f*

_{L}(

*x*) and its Hessian ∇

^{2}

*f*

_{L}(

*x*) are independent of the gradient ∇

*f*

_{L}(

*x*), it will suffice to show that

Fix $x\u2208S2(L)$ and denote by *μ*_{1}(*x*) and *μ*_{2}(*x*) the eigenvalues of the Hessian. First, we show that, conditioned on the event {|*f*(*x*)| ⩽ *α*}, with large probability, |*μ*_{1}(*x*) ⋅ *μ*_{2}(*x*)| = |det ∇^{2}*f*_{L}(*x*)| cannot be too small, and then that, with large probability, max(|*μ*_{1}(*x*)|, |*μ*_{2}(*x*)|) = ‖∇^{2}*f*_{L}(*x*)‖_{op} cannot be too big. Together, these two estimates will do the job.

*X*

_{1},

*X*

_{2}) in the plane Π

_{x}. Then,

*δ*> 0,

*f*

_{L}at

*x*conditioned on

*f*

_{L}(

*x*). The mixed derivative $\u22021,22fL(x)$ 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

*i*= 1, 2, conditioned on

*f*

_{L}(

*x*), is normal with bounded conditional mean

*f*

_{L}is normal and that we are interested only in the values |

*f*

_{L}(

*x*)| ⩽

*α*⩽ 1) and with the bounded conditional variance

*f*(

*x*)| ⩽

*α*}, with probability at least $1\u2212C\delta +e\u2212c\lambda 2$, we have

*λ*=

*δ*Δ and $\delta =12c\Delta \u22121log\Delta $ and noting that $\delta +e\u2212c\lambda 2\u2272\Delta \u221212(log\Delta )14$, 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},

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 $x\u2208S2(L)$ 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

By *p*(*x*, *y*), we denote the probability that two given points $x,y\u2208S2(L)$ 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

*Let*(

*f*

_{L})

*be a regular Gaussian ensemble. Then, the estimates hold uniformly in*

*α*,

*β*⩽ 1

*and in*

*L*⩾

*L*

_{0}

*,*

*with some positive constant*Θ

*, and for*

*d*

_{L}(

*x*,

*y*) ⩾ 1

*, we have*

*p*(

*x*,

*y*) =

*Wp*

^{2}

*, with*

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

#### 1. Beginning the proof

Let $x,y\u2208S2(L)$. 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,

Then,

where

Note that vol(Ω) ≲ *α*^{2}*β*^{4} ≃ *p*^{2}. 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*),

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 *d*_{0}, independent of *L*, so that *λ*(*x*, *y*) is bounded from below by a positive constant whenever *d*_{L}(*x*, *y*) ⩾ *d*_{0}. Hence, proving estimate (8.1), we assume that *d*_{L}(*x*, *y*) ⩽ *d*_{0} [while later, proving the long-distance estimate (8.2), we will assume that *d*_{L}(*x*, *y*) ⩾ *d*_{0}]. The value of sufficiently large constant *d*_{0} is inessential for our purposes.

Denote by $C$ the covariance matrix of ∇*f*_{L}(*x*) and put

Since the matrix $C$ is non-degenerate uniformly in *L* ⩾ *L*_{0} (Lemma 2), the matrix $\Gamma \u0303$ is also non-degenerate uniformly in *L* ⩾ *L*_{0}.

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

Then, provided that *d*_{L}(*x*, *y*) ⩾ *d*_{0} with *d*_{0} ≫ 1, we get

and therefore,

Thus, until the end of the proof of the short-distance estimate (8.1), we assume that *d*_{L}(*x*, *y*) ⩽ *d*_{0} with some positive *d*_{0} independent of *L*.

Let *v*(*x*) denote the three-dimensional Gaussian vector $v(x)=fL(x),\u2207fL(x)t$, and let $a=(\xi 1,\xi 2,\xi 3)t$ and $b=(\xi 4,\xi 5,\xi 6)t$. Then,

whence

By the compactness of the unit sphere in $R3$, there exist $a,b\u2208R3$, |*a*|^{2} + |*b*|^{2} = 1, such that

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

#### 2. The normal and the tangential derivatives

Let $C$ be the big circle on $S2(L)$ that passes through the points *x* and *y*, and let $I\u2282C$ 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

where $v\Vert (x)=fL(x),\u2202\Vert fL(x),0t$ and $v\u22a5(x)=0,0,\u2202\u22a5v(x)t$. Then, by (8.3),

where $a\u2032=(a1,a2,0)t$ and $a\u2032\u2032=(0,0,a3)t$, similarly for *b*′ and *b*^{″}, and |*a*′|^{2} + |*a*^{″}|^{2} + |*b*′|^{2} + |*b*^{″}|^{2} = 1.

Now, consider another Gaussian six-dimensional vector

Since the distribution of *f*_{L} is invariant with respect to orthogonal transformations, the Gaussian vectors *v*(*x*, *y*) and $v\u0303(x,y)$ have the same covariance matrix in the chosen coordinate systems in Π_{x} and Π_{y}. Therefore,

We split the rest of the proof of estimate (8.1) into two cases: (i) $|a\u2032\u2032|2+|b\u2032\u2032|2\u2a7e12$ and (ii) $|a\u2032|2+|b\u2032|2\u2a7e12$. 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): $|a\u2032\u2032|2+|b\u2032\u2032|2\u2a7e12$

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,

Since $|a\u2032\u2032|2+|b\u2032\u2032|2\u2a7e12$, at least one of the following holds:

either $|a\u2032\u2032+b\u2032\u2032|\u2a7e12$ or $|a\u2032\u2032\u2212b\u2032\u2032|\u2a7e12$.

We assume that $|a\u2032\u2032+b\u2032\u2032|\u2a7e12$ (the other case is similar and slightly simpler), let *e* = *a*^{″} + *b*^{″}, $|e|\u2a7e12$, and notice that

We take the point $z\u2208C$, *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,

whence

Put *x*_{0} = *x* and *x*_{1} = *z*, then take the point $x2\u2208C$, *x*_{2}≠*x*_{0}, so that *d*_{L}(*x*_{2}, *x*_{1}) = *d*_{L}(*x*_{1}, *x*_{0}), and continue this way until *d*_{L}(*x*_{0}, *x*_{N}) ⩾ *d*_{0}, where *d*_{0} is the correlation length defined above. Then,

On the other hand, since *d*_{0} is the correlation length and *d*_{L}(*x*_{0}, *x*_{N}) ⩾ *d*_{0}, we have

(at the last step, we use that the distribution of *v*_{⊥} does not degenerate uniformly in *L* ⩾ *L*_{0} and that $|e|\u2a7e12$). Therefore, *λ* ≳ *N*^{−2}.

Recalling that by the definition of *N*, we have (*N* − 1)*d*_{L}(*x*, *y*) < *d*_{0}, and we get *λ* ≳ *d*_{L}(*x*,*y*)^{2}, which concludes our consideration of the first case.

#### 4. Case (ii): $|a\u2032|2+|b\u2032|2\u2a7e12$

In this case, we restrict the function *f*_{L} to the big circle $C$ and treat it as a periodic random Gaussian function $F:R\u2192R$ with translation-invariant distribution. To simplify the notation, we omit the index *L*. By *ρ*, we denote the spectral measure of *F*, that is,

where $X,Y\u2208R$ correspond to the points $x,y\u2208C$. Then, we have

where $\delta =12\pi |X\u2212Y|=12\pi dL(x,y)$. Furthermore,

by the uniform non-degeneracy of ∇*f*_{L} and

by the power decay of correlations of *f*_{L}. Note that since the function *F* is 2*πL*-periodic, the Fourier transform $\rho ^$ of its spectral measure is also 2*πL*-periodic.

Recalling that $12\u2a7d|a\u2032|2+|b\u2032|2\u2a7d1$, we notice that $|a1|+|a2|+|b1|+|b2|\u2a7e12$ 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,

and denote ‖*p*_{δ}‖_{W} = |*a*_{1}| + |*a*_{2}| + |*b*_{1}| + |*b*_{2}| (recall that an exponential sum is the expression

where *λ*_{j} are real numbers and *q*_{j} are polynomials in *ξ* with complex coefficients). Then, the following lemma does the job.

*Let*

*p*

_{δ}

*be the exponential sum*(8.6)

*of degree 4. Let*

*ρ*

*be a probability measure on*$R$

*with the*2

*πL*

*-periodic Fourier transform*$\rho ^$

*. Assume that*

*and*

*Then, given*

*δ*

_{0}> 0

*, there exist*

*c*=

*c*(

*C*,

*γ*,

*m*,

*δ*

_{0}) > 0

*and*

*L*

_{0}=

*L*

_{0}(

*C*,

*γ*,

*m*,

*δ*

_{0})

*such that, for every*0 <

*δ*⩽

*δ*

_{0}

*and every*

*L*⩾

*L*

_{0}

*,*

### C. Proof of Lemma 16

#### 1. Beginning the Proof of Lemma 16

Let $A=2m$. Then, by Chebyshev’s inequality,

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

We claim that

the set Ξ is a union of at most

*B*+ 5 intervals*I*_{j}, 1 ⩽*j*⩽*B*+ 5, andthe length of each interval

*I*_{j}is ⩽*C*(*B*,*δ*_{0})*κ*.

Having these claims, we will show that $\rho [\u2212A,A]\\Xi \u2a7e14$, whence

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 $J\u2282R$ cannot exceed

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

Hence, the set Ξ consists of at most $12((8+2B)+2)=5+B$ 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*,

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

On the other hand,

Thus, |*I*_{j}| ⩽ *C*(*B*, *δ*_{0})*κ*, proving (b).

#### 4. Completing the Proof of Lemma 16

Recall that $\rho ([\u2212A,A])\u2a7e34$ and that given *κ* > 0, we defined the set

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

either $\rho ([\u2212A,A]\\Xi )\u2a7e14$ or $\rho ([\u2212A,A]\u2229\Xi )\u2a7e12$.

In the first case,

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

Suppose that $\rho ([\u2212A,A]\u2229\Xi )\u2a7e12$. Then, $\rho (\Xi )\u2a7e12$. 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

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

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

whence

On the other hand, using the identity

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

Then, recalling that the function $\rho ^$ is 2*πL*-periodic and that $|\rho ^(s)|\u2a7dmin(1,C|s|\u2212\gamma )$ for |*s*| ⩽ *L*/2 and assuming without loss of generality that $\gamma <12$, we get

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

Since the matrix $C$ is non-degenerate uniformly in *L* ⩾ *L*_{0}, the matrix $\Gamma \u0303$ is also non-degenerate uniformly in *L* ⩾ *L*_{0}.

We assume that *d*_{L}(*x*, *y*) ⩾ *d*_{0}, where *d*_{0} is sufficiently large (and independent of *L*). Then, we have

Therefore,

and

Recall that

where

and that

Hence,

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:

*There exist positive* *ɛ*_{0} *and* *c* *and positive* *C* *such that, given* 0 < *ɛ* ⩽ *ɛ*_{0} *and* *L* ⩾ *L*_{0}(*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

and fix a maximal *ρ*-separated set $X(\rho )$ on $S2(L)$. Then, $|X(\rho )|\u2243(L/\rho )2$.

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 $\Vert fL\Vert C3<log\u2061L$ and with the w.h.p.-estimate $maxCr(\alpha )\Vert (\u22072fL)\u22121\Vert op\u2a7dL3\epsilon $ provided by Lemma 13. Hence, we need to estimate the probability of the event

with an appropriately chosen constant *C*.

Suppose that the event $E$ occurs. Denote by *x*_{1} and *x*_{2} the closest to *z*_{1} and *z*_{2} points in $X(\rho )$. Then,

and

i.e., *x*_{1}, *x*_{2} ∈ Cr(2*α*, *β*). We claim that

*the mean number of pairs of points*$x1,x2\u2208Cr(2\alpha ,\beta )\u22c2X(\rho )$*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

where $p(x1,x2)=P{x1,x2\u2208Cr(2\alpha ,\beta )}$ is the two-point function estimated in Lemma 15. By Lemma 15, *p*(*x*_{1}, *x*_{2}) ≲ *L*^{4ɛΘ}*p*^{2}, so the whole sum is

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

We fix a maximal *r*-separated set $X(r)$ on $S2(L)$. Then, the disks *D*(*x*, *r*), $x\u2208X(r)$, cover $S2(L)$ with a bounded multiplicity of covering and $|X(r)|\u2243(L/r)2$. We set

W.o.p., for *L* ⩾ *L*_{0}, we have $\Vert fL\Vert C3\u2a7dlog\u2061L$. Then, by Lemma 4, each disk *D*(*y*, *r*), *y* ∈ *Y*, contains a unique critical point *z* ∈ Cr(*α*), and

provided that *L* ⩾ *L*_{0}. This yields two useful observations that hold w.o.p.:

|Cr(

*α*)|≳|*Y*|, andif

*y*_{1},*y*_{2}∈*Y*, then either*d*_{L}(*y*_{1},*y*_{2}) ⩽ 2*r*and the number of such pairs (*y*_{1},*y*_{2}) is $\u2272|X(r)|$ or*d*(*y*_{1},*y*_{2}) ⩾*L*^{−ɛ}. Indeed, if the points*y*_{1},*y*_{2}∈*Y*generate the same critical point*z*, then*d*_{L}(*y*_{1},*y*_{2}) ⩽ 2*r*. 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 $\Vert (\u22072fL(z))\u22121\Vert op\u2a7d\Delta $ is*c*Δ^{−1}(log*L*)^{−1}-separated, and thus, in this case,*d*(*y*_{1},*y*_{2}) ⩾*L*^{−ɛ}.

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

and that

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

#### 1. Estimating $E[|Y|]$

This estimate is straightforward,

#### 2. Estimating Var**[**|*Y*|**]**

In this section, *p* and *p*(*x*, *y*) will denote the one- and two-point functions of the set $Cr13\alpha ,\beta $. Given $x,y\u2208S2(L)$, we put

Then, $E[|Y|]=p\Delta |X(r)|$ and

The first sum on the RHS is bounded by

Hence, we need to estimate the double sum only.

In the double sum, we consider separately the terms with *d*_{L}(*x*, *y*) ⩽ 2*r*, the terms with 2*r* < *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*) ⩽ 2*r*.

Taking into account that the number of such pairs is $\u2272|X(r)|$, we bound this sum by $\u2272p\Delta |X(r)|=E[|Y|]\u2272(9.3)L\u221212\epsilon (E[|Y|])2$.

##### b. The terms with 2*r* < *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 $x,y\u2208Cr(13\alpha ,\beta ,12\Delta )$ with 2*r* < *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 $|X(r)|2\u2243(L/r)4\u226a(L/\beta )4\u226aL10$, 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

Then, the whole double sum is

which gives what we needed with a large margin.

##### d. The terms with *d*_{L}(*x*, *y*) ⩾ *L*^{ɛ}.

In this case,

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

while, by Lemma 14,

Thus, $p\Delta (x,y)\u2212p\Delta 2\u2272L\u2212c\epsilon p\Delta 2$, and the whole double sum is bounded by $L\u2212c\epsilon p\Delta 2\u22c5|X(r)|2=L\u2212c\epsilon E[|Y|]2$.

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

## X. ASYMPTOTIC INDEPENDENCE

*Let*(

*f*

_{L})

*be a regular Gaussian ensemble, and let*$Z\u2282S2(L)$

*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*

*c*

_{1},

*c*

_{2}

*so that, for*

*L*⩾

*L*

_{0}

*,*

*L*and introduce the following notation:

$HfL$ is a Gaussian Hilbert space generated by

*f*_{L}, i.e., the closure of finite linear combinations*∑c*_{j}*f*_{L}(*x*_{j}) with the scalar product generated by the covariance.$H$ is “a big Gaussian Hilbert space” that contains $HfL$ and countably many mutually orthogonal one-dimensional subspaces that are orthogonal to $HfL$.

*J*(*z*) =*f*_{L}(*z*),*z*∈*Z*, are unit vectors in $HfL$ with

*there exists a collection of orthonormal vectors*$J\u0303(z)z\u2208Z\u2282H$

*such that*

*z*,

*z*′)

_{z,z′∈Z}with the elements

*L*>

*L*

_{0}(

*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

*z*∈

*Z*,

*t*(

*z*) < Γ(

*z*,

*z*), which is easy to see since

*κ*is sufficiently small. Since the Hermitian matrix Γ is positive-definite, we can find a collection of vectors ${I(z)}z\u2208Z\u2282H\u2296span{J(z)}z\u2208Z$ with the Gram matrix Γ. Note that $\Vert I(z)\Vert H=\Gamma (z,z)=L\u22120.45\gamma $. Then, we let

*z*∈

*Z*, we have

*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

*Given*$0<p0\u2a7d12$

*, 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}

*,*

*p*

_{0}⩽

*p*

_{j}⩽ 1 −

*p*

_{0}

*, and the value 0 with probability*1 −

*p*

_{j}

*. Let*

*Then, there exists*

*ɛ*=

*ɛ*(

*p*

_{0}) > 0

*such that for any measurable function,*

*Q*: Ω → [0, 1]

*with*$\u222b\Omega QdP\u2a7e1\u2212\epsilon $

*, and for any*$m\u2208R$

*, we have*

*c*(

*p*

_{0}) may vary from line to line. Take $\lambda =1/N$. First, we claim that $|E[ei\lambda SN]|\u2a7d1\u2212c(p0)$. Indeed,

*e*

^{it}− 1| ⩽ |

*t*|, we proceed as follows:

*ɛ*⩽

*c*(

*p*

_{0})/4, we complete the proof.□

### B. Large sections lemma

*Let*$(\Omega 1\xd7\Omega 2,P1\xd7P2)$

*be a product probability space, and let*0 <

*p*⩽ 1

*and*$0<\epsilon \u2a7d12p$

*. Let*

*Q*: Ω

_{1}× Ω

_{2}→ [0, 1]

*be a measurable function with*$\u222b\Omega 1\xd7\Omega 2QdP\u2a7e1\u2212\epsilon $

*, and let*

*X*⊂ Ω

_{1}

*be an event with*$P1(X)\u2a7ep$

*. Then,*

In the following, we will apply this lemma, mostly, with *X* = Ω_{1} and *p* = 1.

## XII. VARIANCE OF THE NUMBER OF LOOPS

Let *G* = *G*(*V*, *E*) be a finite graph embedded in the sphere $S2$ 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 $S2$ connecting the vertices, and the faces are the connected components of the open set $S2\(V\u222aE)$.

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

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 $(\Omega ,P)$, we denote the probability space on which the random states are defined. Then, *N*(Γ) = *N*(Γ(*σ*_{V})) is a random variable on $(\Omega ,P)$.

Given $0<p0\u2a7d12$, we put $V(p0)=v\u2208V:p0\u2a7dp(v)\u2a7d1\u2212p0$ and denote by |*V*(*p*_{0})| the cardinality of the set of vertices *V*(*p*_{0}).

*For any*$0<p0\u2a7d12$

*, there exist positive*

*c*(

*p*

_{0})

*,*

*C*(

*p*

_{0})

*, and*

*ɛ*=

*ɛ*(

*p*

_{0})

*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*$\u222b\Omega Q(\sigma V)dP\u2a7e1\u2212\epsilon $

*and for any*$m\u2208R$

*,*

*provided that*|

*V*(

*p*

_{0})| ⩾

*C*(

*p*

_{0})

*.*

### 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 $\sigma V=(\sigma V\u2032,\sigma V\u2032\u2032)$. A collection of states *σ*_{V′} assigned to the vertices from *V*′ generates (Fig. 5)

a collection Γ′ = Γ(

*σ*_{V′}) of disjoint loopsand a graph

*G*(*σ*_{V′}) with vertices at*V*^{″}, all of them having degree 4.

By *N*(Γ′), we denote the number of loops in the collection Γ′. The collection of states $\sigma V\u2032\u2032$ turns the graph *G*(*σ*_{V′}) into a collection of loops $\Gamma \u2032\u2032=\Gamma (\sigma V\u2032,\sigma V\u2032\u2032)$. 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 $\sigma V\u2032\u2032$ are independent, we have

Hence, for any event *X*′ ⊂Ω,

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

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,

close to 1, while $P(X\u2032)$ 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 $\sigma V\u2032\u2032$ 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*) < *p*_{0} or *p*(*v*) > 1 − *p*_{0}

As above, we let $V(p0)=v\u2208V:p0\u2a7dp(v)\u2a7d1\u2212p0$. We put *V*′ = *V*\*V*(*p*_{0}) and *V*^{″} = *V*(*p*_{0}) and consider

Then, by Lemma 20 (applied with *p* = 1), $P(X\u2032)\u2a7e12$.

Hence, from now on, we assume that for each vertex *v* of the graph *G*, we have *p*_{0} ⩽ *p*(*v*) ⩽ 1 − *p*_{0}, while $\u222b\Omega Q(\sigma V)dP\u2a7e1\u22122\epsilon $.

### 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 $f\u2208F$, we denote by $v(f)$ the set of vertices in *V* that lie on the boundary $\u2202f$. The Euler formula gives us

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

We let $F*={f\u2208F:|v(f)|\u2a7d4}$ be the set of all faces having at most 4 vertices on the boundary and conclude that $F*>15|F|>15|V|$.

### D. Choosing a maximal collection of separated faces

We call faces $f,f\u2032\u2208F*$ *separated* if they do not have common vertices on their boundaries: $v(f)\u2229v(f\u2032)=\u2205$. We fix a maximal collection $F\u2282F*$ of separated faces. Since for any face $f\u2208F$, there are at most 12 other faces $f\u2032\u2208F*$ with $v(f\u2032)\u2229v(f)\u2260\u2205$, we conclude from the maximality of $F$ that it also contains sufficiently many faces,

### E. Marking vertices and cycles

Next, we choose a simple cycle $c$ on the boundary of each face $f\u2208F$ and mark a vertex $vc$ on that cycle according to the following rule: We take a vertex *v*′ in $v(f)$ and, starting at *v*′, walk along $\u2202f$ 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).

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.

### 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 $vc$ with itself (Fig. 7). Note that this event depends only on the states of at most 3 unmarked vertices lying on $c\{vc}$. Therefore, for any marked cycle $c$, we have $Pcis\u2009good\u2009\u2a7ep03$. Hence, denoting by *N*_{G} the number of good cycles, we obtain

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

To simplify the notation, we let $d(p0)=1130p03$ and $D(p0)=d(p0)\u22122$. Then, letting $X=\omega \u2032:NG>d(p0)|V|$, we see that $P(X)\u2a7ep=def1\u2212D(p0)|V|\u22121$. Put

(recall that at this moment $\u222b\Omega QdP\u2a7e1\u22122\epsilon $). Then, by Lemma 20, $P(X\u2032)\u2a7e12p$, which is $\u2a7e14$, provided that |*V*| > 2*D*(*p*_{0}). 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, $\u222b\Omega QdP\u2a7e1\u22124\epsilon p\u22121$.

### 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

By Lemma 20 (applied with *X* = Ω), $P(X\u2032)\u2a7e12$. 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 $\sigma VM.G.*$ of good marked vertices such that that none of the corresponding good cycles turns into a separate loop and denote by $\Gamma (\sigma VM.G.*)$ the loop ensemble obtained from the graph *G* after the assignment of the states $\sigma VM.G.*$. Introduce a collection of independent Bernoulli random variables $\eta vv\u2208VM.G.$, letting *η*_{v} = 0 if $\sigma v=\sigma VM.G.*(v)$ and *η*_{v} = 1 otherwise. Then,

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

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*(*p*_{0})|*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

This is a random Gaussian function equidistributed with *f*_{L}. We will show that

which immediately yields the lower bound for Var[*N*(*f*_{L})] we are after. Note that

That is, it suffices to show that with probability at least $12$ in *f*_{L}, we have

### 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}∈*C*^{3}(*A*, Δ,*α*,*β*) (introduced in Sec. VI A) with*A*= log*L*, Δ =*L*^{3ɛ}, and with*β*chosen so that*β*^{2}*L*^{7ɛ}=*α*(i.e., $\beta =L\u22121\u221252\epsilon $),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 $P=PgL$, $E=EgL$, 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

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

which holds w.h.p. with a large margin since *A*Δ^{2}*α*^{2} = *L*^{−4+10ɛ} log *L*, while, as we will momentarily see, w.h.p. in *g*_{L}, we have

where *c*_{1} ⩽ 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

at a given point *p* is $\u2272L\u2212c1$. By the union bound, the probability that this happens somewhere on Cr(*α*) is

provided that *ɛ* is sufficiently small.

Thus, Lemma 12 applied to the functions *f*_{L} and $f\u0303L$ yields that

on the major part of the probability space where *g*_{L} ∈ *C*^{3}(log *L*) and where estimate (13.1) holds. The first term on the RHS, $NI(f\u0303L)$, comes from the stable connected components of *Z*(*f*_{L}). Hence, on the large part of the probability space, the fluctuations in $N(f\u0303L)$ come only from the blinking circles $NII(f\u0303L)$ and from the Bogomolny–Schmit loops $NIII(f\u0303L)$, and after we have fixed the function *f*_{L}, both these quantities depend only on the configuration of (random) signs of $f\u0303L(p)$ 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

Denote by Ω′ the event that $\Vert gL\Vert C3\u2a7dlog\u2061L$, and both estimates (13.1) and (13.2) hold. Then,

while on Ω′, we have

Note that the random signs *s*(*p*) are independent and that there exists $p0\u2208(0,12]$ such that on Cr(*α*′) each of the two possible values of *s*(*p*) is attained with probability at least *p*_{0}.

For any subset *Z* ⊂ Cr(*α*), we let $sZ=s(p)p\u2208Z$. As before, we use the notation

Since, on Ω′, $NII(f\u0303L)$ depends only on $sCrE(\alpha )$ and $NIII(f\u0303L)$ on $sCrS(\alpha )$, there exist functions $N\u0303II(sCrE(\alpha ))$ and $N\u0303III(sCrS(\alpha ))$ such that, on Ω′, we have $NII(f\u0303L)=N\u0303II(sCrE(\alpha ))$ and $NIII(f\u0303L)=N\u0303III(sCrS(\alpha ))$. The function $NI(f\u0303L)$ stays constant on Ω′, and by $N\u0303I$, we denote the value of that constant. Denoting by *χ*_{Ω′} the indicator function of the event Ω′, we get

The conditional expectation $E[\chi \Omega \u2032sCr(\alpha )]$ 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,

Note that $E[Q]=P(\Omega \u2032)=1\u2212\tau (L)$.

### E. Fluctuations generated by the Bogomolny–Schmit loops

First, we consider the case when $|CrS(\alpha \u2032)|\u2a7e12|Cr(\alpha \u2032)|$ 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 $sCr(\alpha )=(sCrE(\alpha ),sCrS(\alpha ))$ and let

Then, by Lemma 20, $P(X\u2032)\u2a7e12$, and therefore,

We fix *ω*_{1} ∈ *X*′ and the corresponding signs $sCrE(\alpha )(\omega 1)$ 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*(*A*Δ)^{−1} with sufficiently small positive constant *c*. The edges are connected components of the set

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 [*p*_{0}, 1 − *p*_{0}]. Then, Lemma 21 yields that

This finishes the proof of our theorem in the case when $|CrS(\alpha \u2032)|\u2a7e12|Cr(\alpha \u2032)|$.

### 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

Then, by Lemma 20, $P(X\u2032)\u2a7e12$. We fix *ω*_{1} ∈ *X*′ and the corresponding $sCr(\alpha )\CrE(\alpha \u2032)(\omega 1)$. The value of $N\u0303II$ is the number of *p* ∈ Cr_{E}(*α*′) such that the sign *s*(*p*) is opposite to the sign of the eigenvalues of $HfL(p)$. To each *p* ∈ Cr_{E}(*α*′), we associate a Bernoulli random variable

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 ⩾*p*_{0}, and note that

Then, Lemma 19 does the job. This finishes off the proof of our theorem in the second case when $|CrE(\alpha \u2032)|\u2a7e12|Cr(\alpha \u2032)|$.□

## 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 $RP2$ 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 $S2(n)$.

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 (*p*_{1}, −*p*_{1}) and (*p*_{2}, −*p*_{2}), the random variables *f*_{n}(*p*_{1}) and *f*_{n}(*p*_{2}) 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 $S2(n)$ 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*(*A*Δ)^{−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 $Hfn(p)$, *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 $f\u0303n$ remain positive on *∂D*(*p*, *δ*) and convex in *D*(*p*, *δ*). By Lemma 5, $f\u0303n$ has a critical point *p*_{t} ∈ *D*(*p*, *δ*), which is its local minimum. If the blinking circle occurs, when it disappears, $f\u0303n$ 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

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 $\u2212f$ have a common vertex *v* (the case in which −*v* will be also a common vertex of $f$ and $\u2212f$). This follows from the following lemma.

*Let* $X,X\u2032\u2282S2$ *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 $\u2212f$ 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 $\gamma \{v,\u2212v}\u2282f$ and by the path −*γ* with $\u2212\gamma \{v,\u2212v}\u2282\u2212f$. Put *X* = *γ* ∪ −*γ*. This is a symmetric closed connected subset of $S2$. If $f\u2032$ is another such face (different from $\xb1f$), then we have another symmetric closed connected set *X*′, and by Lemma 22, *X* ∩ *X*′ ≠ *∅*. Since (*γ* ∪ −*γ*) \{*v*, −*v*} is contained inside $f\u222a\u2212f$, while $(f\u222a\u2212f)\u2229(f\u2032\u222a\u2212f\u2032)=\u2205$, we see that *v* and −*v* must be vertices on $\u2202f\u2032\u222a\u2202(\u2212f\u2032)$ as well. Recalling that each vertex on our graph has degree 4, we see that there are at most eight such “bad faces” $f$.

Assume that *X*_{1} ∩ *X*_{2} = *∅*. Without loss of generality, we assume that the set *X*_{2} contains the North and the South Poles. Since *X*_{1} and *X*_{2} are compact, there exists *ɛ* > 0 such that dist(*X*_{1}, *X*_{2}) > 6*ɛ*.

Take any point *z* ∈ *X*_{1}. Then, −*z* ∈ *X*_{1} as well. Since *X*_{1} is connected, there exists a finite chain of points in *X*_{1} *z* = *z*_{0}, *z*_{1}, …, *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 *γ*_{1} joining *z* to −*z* and staying in the *ɛ*-neighborhood of *X*_{1}. Let *γ* = *γ*_{1} ∪ (−*γ*_{1}). Then, *γ* is a symmetric curve going from *z* to −*z* and back and staying in the *ɛ*-neighborhood of *X*_{1}.

In a similar way, we can construct a curve from the North Pole to the South Pole staying in the *ɛ*-neighborhood of *X*_{2}. Let *γ*_{2} 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 *γ*_{2} and both these circles stay in the *ɛ*-neighborhood of *X*_{2} and, thereby, are disjoint with *γ*.

*x*

_{3}= ±cos

*ɛ*} of

*C*, while $\gamma \u03032$ joins those edge circles.

Consider the universal covering map $p:S\u2192C$, where $S={(t,s)\u2208R2:|s|\u2a7dcos\u2061\epsilon}$ is a horizontal infinite strip and $p((t,s))=(cos\u20612\u2061\pi t,sin\u20612\u2061\pi t,s)$. Note that, for $\u2113\u2208Z$, $p((t+12+\u2113,\u2212s))=\u2212p((t,s))$. By the path lifting lemma, $\gamma \u03032$ is lifted to some curve Γ_{2} on *S* joining the top and the bottom boundary lines. The curve $\gamma \u03031$ is lifted to some curve Γ_{1}(*τ*), *τ* ∈ [0, 1], joining (*t*_{0}, *s*_{0}) with (*t*_{0} + *ξ*, −*s*_{0}), where $\xi \u2208Z+12$. Then, the curve $\Gamma 1*(\tau )=(t(\tau )+\xi ,\u2212s(\tau ))$ extends Γ_{1} and projects to $\u2212\gamma \u03031$. 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 ∩Γ_{2} = *∅*, so the increment of arg(*w* − *w*_{2}), as *w* runs over Γ and *w*_{2} ∈ Γ_{2} stays fixed, should not depend on *w*_{2}. However, this increment is +*π* when *w*_{2} is on the top boundary line of *S* and −*π* when *w*_{2} 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.)