Critical point asymptotics for Gaussian random waves with densities of any Sobolev regularity

We consider Gaussian random monochromatic waves $u$ on the plane depending on a real parameter $s$ that is directly related to the regularity of its Fourier transform. Specifically, the Fourier transform of $u$ is $f\,d\sigma$, where $d\sigma$ is the Hausdorff measure on the unit circle and the density $f$ is a function on the circle that, roughly speaking, has exactly $s-\frac12$ derivatives in $L^2$ almost surely. When $s=0$, one recovers the classical setting for random waves with a translation-invariant covariance-kernel. The main thrust of this paper is to explore the connection between the regularity parameter $s$ and the asymptotic behavior of the number $N(\nabla u,R)$ of critical points that are contained in the disk of radius $R\gg1$. More precisely, we show that the expectation $\mathbb{E}N(\nabla u,R)$ grows like the area of the disk when the regularity is low enough ($s<\frac32$) and like the diameter when the regularity is high enough ($s>\frac52$), and that the corresponding exponent changes according to a linear interpolation law in the intermediate regime. The transitions occurring at the endpoint cases involve the square root of the logarithm of the radius. Interestingly, the highest asymptotic growth rate occurs only in the classical translation-invariant setting, $s=0$. A key step of the proof of this result is the obtention of precise asymptotic expansions for certain Neumann series of Bessel functions. When the regularity parameter is $s>5$, we show that in fact $N(\nabla u,R)$ grows like the diameter with probability 1, albeit the ratio is not a universal constant but a random variable.


Introduction
Nazarov and Sodin have developed some powerful techniques to derive asymptotic laws for the distribution of the zero set of smooth Gaussian functions of several variables [16,17].Specifically, their theory applies to two different but related settings: the restriction to large balls of Gaussian functions on Euclidean space with translation-invariant covariance kernels and to Gaussian ensembles of high degree polynomials on the sphere or the torus with asymptotically translation-invariant kernels.In the first setting, a prime example arising in spectral theory is the study of Gaussian random monochromatic waves; in the second, that of random spherical harmonics of high frequency.
In this paper we are concerned with asymptotic laws for the number of critical points (i.e., the zeros of the gradient).We consider this question in the context of Gaussian random monochromatic waves on the plane, which are solutions to the Helmholtz equation on R 2 , ∆u + u = 0 . (1.1) As is well known, the study of critical points is a central topic in spectral theory [21,22,14,6] (and, in general, in the geometric study of solutions to differential equations [20,1,2,9]), both in the deterministic and random settings.This is partly because they are very closely related to the geometry of the nodal components.When u is polynomially bounded, the Helmholtz equation simply means that u is the Fourier transform of a distribution supported on the unit circle, which we identify with T := R/2πZ via the map E(φ) := (cos φ, sin φ) . (1.2) Solutions to the Helmholtz equation are necessarily analytic, but their Fourier transforms do not have any a priori regularity properties.There are some connections, though, between the regularity of the Fourier transform and the decay rate of u at infinity.Most important is the classical result of Herglotz ensuring that u has the sharp fall-off at infinity (which is as |x| − 1 2 in a space-averaged sense) if and only if one can write with some square-integrable density f , and that in this case the norm f L 2 (T) quantitatively captures the decay rate of u.For details and generalizations, see e.g.[10,Appendix A].
The main thrust of this paper is to understand the connection between the distribution of the critical points of u, defined as in (1.3), and the regularity of the density f .To this end, we consider the usual ansatz for random plane waves [7,19] and tweak it by introducing a real parameter s ∈ R to control the regularity of f : a l |l| −s e ilθ J l (r) . (1.4) Here the real and imaginary part of a l are independent standard Gaussian random variables subject to the constraint a l = (−1) l a −l (which makes u real valued), (r, θ) ∈ R + × T are the polar coordinates.This is equivalent to taking the Gaussian random density and then defining u through the formula (1.3), which must be understood in the sense of distributions.
Of course, the rationale behind this definition is that {|l| −s e ilφ } l =0 is an orthonormal basis of the Sobolev space Ḣs (T) of functions with zero mean and s derivatives in L 2 , which reduces to the space of square-integrable functions of zero mean when s = 0.The covariance kernel of u is translation-invariant when s = 0, so the Nazarov-Sodin theory is applicable in this case (see Remark 4.2 for details), but this is not the case for nonzero s.One should note that the proofs work verbatim if one replaces the weight |l| −s by a more general expression such as σ l = σ −l = |l| −s + p −s−1 (l) , (1.6) where the function p −s−1 (t) is an arbitrary classical symbol of order −s − 1 (which does not necessarily vanish at 0).The resulting constants, however, depend on the specific sequence σ l .
It is not hard to see that the parameter s describes the regularity of the density in the sense that f has exactly s − 1 2 derivatives in L 2 almost surely, as measured using Sobolev or Besov spaces.Specifically, one can show that, for any δ > 0, with probability 1; see Proposition 2.2 for details.
Our main result provides an asymptotic estimate for the growth of the expected number of critical points contained in a disk of large radius R, which we denote by N (∇u, R) := #{x ∈ B R : ∇u(x) = 0} , as a function of the regularity parameter s.It is elementary that this quantity is an upper bound for the expected number of nodal components contained in B R .With the usual ansatz for random plane waves, it is well known that N (∇u, R) grows asymptotically like the area of the disk; more precisely [4], when s = 0 one has where κ(0) := 1/(2 √ 3) and where the notation q(R) ∼ Q(R) means that the quotient q(R)/Q(R) tends to 1 as R → ∞.
We should mention from the onset that the effect of changing the regularity parameter s can be quite drastic, as one should not expect that the number of critical points grows like the area in all regularity regimes.To illustrate this, recall that, when s = 0, the Nazarov-Sodin theory ensures the number of nodal components of u contained in B R grows as almost surely for some constant ν 0 > 0. In contrast, the results proven in [10] show that almost surely for s > 4, with ν ∞ := 1/π.Understanding the asymptotic behavior of the number of nodal components in other regimes is an extremely challenging open problem.Consequently, our main objective in this paper is to analyze the intriguing transitions between distinct asymptotic regimes in the simpler case of critical points.
In the case of critical points, it is also natural to wonder about the asymptotic growth in the case of very negative regularities s < 0. Recall that, by the Faber-Krahn inequality, the number of nodal components of a solution to the Helmholtz equation contained in B R is at most cR 2 , where c is a universal constant.However, the number of critical points is not bounded a priori: in Appendix A we show that, given any continuous function ρ : R + → R + , there exists a solution to the Helmholtz equation on R 2 having at least ρ(R) nondegenerate critical points in B R , for all R > 1.Thus, one could in principle expect the average number of critical points in a large ball R to have a fast growth in R for small enough regularities.
Our main result provides a satisfactory, and quite surprising, answer to both questions.It turns out that the growth of the expected number of critical points is like the square of the radius for s < 3  2 , linear for s > 5  2 , and the corresponding exponent changes according to a linear interpolation law in the intermediate regime 3  2 < s < 5 2 .The transitions occurring at the endpoint cases involve not only a power law, but also the square root of the logarithm of the radius.Furthermore, the highest asymptotic growth of the expected number of critical points is attained exactly for s = 0, that is, in the usual setting of random plane waves.
(ii) In the region where the growth of EN (∇u, R) is volumetric, the constant κ(s) depends continuously on s.More precisely, it is only Lipschitz at s = 1 2 .Furthermore, κ(s) is strictly increasing on (−∞, 0), strictly decreasing on (0, 3  2 ), and tends to 0 as s → −∞ and as s → Figure 1 summarizes Theorem 1.1 in a more visual way.The fact that the highest asymptotic growth for the number of critical points occurs precisely in the translation-invariant case s = 0 is somewhat surprising.Naively one could expect that rougher density functions, which feature wilder oscillations, would exhibit more critical points.Theorem 1.1 shows that, strictly speaking, this is only the case for regularities s > 0.
Let us now discuss the proof of Theorem 1.1.The asymptotic analysis of N (∇u, R) hinges on the celebrated Kac-Rice counting formula, which, under suitable technical hypotheses, expresses the expected number of zeros of a random field (in this case, the gradient ∇u) has in terms of a multivariate integral.As is well known, this formula has been used profusely in the literature [11,17,4,5], and in particular lies at the heart of the computation of EN (∇u, R) for s = 0 and of the finer asymptotics bounds for the expected number of extrema and saddle points and for higher order correlations obtained in [4] also in the translation-invariant case s = 0.
The coefficients that appear in the Kac-Rice integral formula involve, via the variance matrix of ∇u, weighted series of Bessel functions of the form where m and m are certain integers.J s,m,m is sometimes called in the literature a second type Neumann series.It is clear that the way each term J l+m (r) J l+m (r) contributes to the sum for r 1 and l 1 will depend on whether the "angular frequency" l is much larger than r, much smaller than r, or roughly of the same size; moreover, the effect of each group of angular frequencies will have a different relative weight in the sum depending on the power s appearing in l −2s .More precisely, a key step of the proof is to establish the following technical result, which controls the asymptotic behavior of J s,m,m (r): Lemma 1.2.For any pair of nonnegative integers m, m and any real s, the large-r asymptotic behavior of J s,m,m is with some explicit constants that will be defined later on.
Ultimately, the different asymptotic regimes that the expectation of N (∇u, R) can exhibit can be traced back to the asymptotic behavior of functions of the form (1.7).One should note that, in general, the highly oscillatory nature of summands in (1.7) makes the analysis of the asymptotic behavior of J s,m,m (r) rather subtle.An exception to this general fact is precisely the case s = 0, where all the associated series can be computed exactly using that the covariance kernel of u is translation-invariant (or, equivalently, the addition formula for Bessel functions); this makes it much easier to analyze the corresponding asymptotic behavior of EN (∇u, R).To illustrate this fact, in the very short Appendix B we carry out the analysis of the translation invariant case s = 0.
In the particular case of smooth enough density functions, one can use the methods of our previous paper [10] to understand the asymptotic behavior of the number of critical points (not only of its expectation value) in greater detail.Specifically, one can prove the following: with probability 1.In particular, N (∇u, R) grows linearly almost surely.
Here the random variable (which is at least 2 almost surely) denotes the number of critical points of the (non-Gaussian) random function |f |.In particular, the asymptotic growth of N (∇u, R) is linear with probability 1, albeit the ratio is not a universal constant but a random variable.In view of Theorem 1.1, a consequence of this asymptotic formula is an explicit formula for the expectation EN (|f | ) when s > 5.
The paper is organized as follows.In Section 2, we start by showing the relation between the parameter s and the regularity of the random function u.Sections 3, 4 and 5 are respectively devoted to the proofs of Lemma 1.2 and Theorems 1.1 and 1.3.We have divided each of these sections into a number of subsections to emphasize the main ideas of each proof.The paper concludes with two Appendices.In Appendix A, we construct solutions to the Helmholtz equation on the plane for which the number of nondegenerate critical points contained in B R grows arbitrarily fast as R → ∞.In Appendix B, we revisit the translation-invariant case (s = 0) and explain the key simplifications that appear in this extremely important case.

Almost sure regularity of the random density function
Our objective in the section is to show that, with probability 1, the Gaussian random function f , defined in (1.5), has exactly s − 1 2 derivatives in L 2 , measured using suitable Sobolev or Besov spaces.
To prove the main result we will need the following version of the strong law of large numbers for sequences of random variables that are labeled by two integers: N =1 be a sequence of positive integers such that lim inf Proof.The strong law of large numbers ensures that converges to 0 almost surely as M → ∞, with Q M := M N =1 K N .Thus, from the identity almost surely.Notice that we have used the assumption lim inf M →∞ The lemma then follows.
We are now ready to prove the main result of this section.Here and in what follows, we shall use the notation q ≈ Q or q Q when there exists a constant C (independent of the large parameter under consideration) such that Q/C q CQ or q CQ, respectively.Proposition 2.2.For each δ > 0, the Gaussian random function (1.5) satisfies Proof.Let us recall that the H σ (T) norm of the function f defined in (1.5) is To analyze this quantity, consider the set of integers Λ N := {l : 2 N −1 l < 2 N } and the subsequences is bounded away from zero.Hence one can apply Lemma 2.1 to infer that almost surely as N → ∞.Therefore, with probability 1, This shows that, with probability 1, f H σ (T) < ∞ if and only if σ < s − 1 2 .The estimate for the Besov norm follows from an analogous reasoning using that Remark 2.3.The result and the proof remain valid in higher dimensions with minor modifications.Specifically, let {Y lm : 1 m d l , 0 l < ∞} be an orthonormal basis of spherical harmonics on the unit (n − 1)-dimensional sphere S n−1 , with ∆ S n−1 Y lm + l(l + n − 2)Y lm = 0. Consider the Gaussian random function where a lm are independent standard Gaussian variables and s ∈ R. Then almost surely.
To spell out the details, the proof in higher dimension starts with the formula 1) .Lemma 2.1 then ensures converges to 1 almost surely as N → ∞, and the result follows from the same argument as above.Obviously, the result also remains valid if one replaces the weight l s by another quantity w l ≈ l s .

Asymptotics for weighted Bessel series
In this section we shall prove Lemma 1.2.In view of the well-known asymptotics for Bessel functions, it is easy to check that the series is locally uniformly convergent by the standard bound [18, (10.14.4)] We are interested in the effect of the parameters s ∈ R and m , m ∈ Z.
In view of the well-known integral representation formula [18, (10.9. 2)] for Bessel functions of integer order, Notice that we have used that J l is real valued, and hence J l = J l .A straightforward application of the stationary phase formula [13, Theorem 7.7.5]gives the following asymptotic formula for g λ .Here and in what follows, we will use the notation Also, we will use the notation O p (r −k ) to emphasize that a certain quantity of order r −k is not bounded uniformly with respect to the parameter p.
Lemma 3.1.Suppose that λ = 1.For r 1, one then has where the error term is not bounded uniformy for large λ or for λ close to 1.
Therefore, the asymptotic analysis of g λ (r) becomes problematic when λ is close to 1 (because in this case the phase function presents degenerate or "almost degenerate" critical points) and when λ is large (because the error terms are not uniformly bounded in this case).Consequently, we will fix a small parameter δ > 0 and consider smooth cutoff functions We can then split J s,m,m (r) as Note that I only involves frequencies smaller than (1 − δ)r, II involves frequencies close to 1 (more precisely, in the interval (1 − 2δ)r < l < (1 + 2δ)r), and III involves frequencies larger than (1 + δ)r.

3.1.
The small frequency region.In view of the asymptotic expansion for g λ (r) proved in Lemma 3.1, it is natural to consider the closely related quantities Let us start by analyzing the large r behavior of I when s 1 2 : Lemma 3.2.For r 1 and some η > 0 depending on s, ) if s = 1 2 and ν is odd.
Proof.Let us start with the case s 0. The basic observation here is that, as the function by standard results about the convergence of Riemann sums for integrands of bounded variation.If s 0, the result then follows from the formula and the estimate arcsin 1 For s ∈ (0, 1 2 ), the integrand is an unbounded function in L 1 loc , so the argument does not apply.Let us take a small constant ε such that, for simplicity of notation, εr is an integer, and write Obviously, as |h(λ)| ≈ λ −2s for small λ, and , we conclude that To estimate I 2 , we use that where the constant in depends on δ.
Putting together the estimates for I 1 and I 2 with ε ≈ r − 1 2 , we obtain Using again the formula (3.4), this proves the lemma when s ∈ (0, 1 2 ).
Let us now pass to the case s = 1 2 .We start by assuming that the integer ν is odd, so that cos πν it turns out that the corresponding integrand is differentiable at λ = 0 in this case, so the same arguments as in the case s < 0 show The result then follows from the formula To conclude, consider the case when s = 1 2 and ν is even.Obviously, by (3.5), 4π r where the constant in depends on δ.The leading contribution of this sum is therefore given by the harmonic series, which satisfies This completes the proof of the lemma.Now we pass to analyzing the contribution of the second term, I .As this term is somewhat oscillating due to the presence of the large parameter r in the argument of a sine, it makes sense to expect this term should be subdominant.

Lemma 3.3.
There exists some η > 0, depending on s, such that Proof.We start with the case s < 1 2 .Let β ∈ (0, 1) be some constant that we will specify later and write , the first term can be easily estimated as By hypothesis, the RHS is r −η for some η > 0.
To estimate I 2 , decompose the interval ( r β , (1 − δ)r] as the union of N disjoint intervals of the form (l n , l n + Λ n ].We assume that l n are integers and that the lengths of the intervals satisfy Λ n ≈ r γ for some γ ∈ (0, β).This implies that N ≈ r 1−γ .
The basic idea is that, with this choice of the scales, one can expect that the function h will be approximately constant in each interval but the phase of the complex exponential will oscillate rapidly.This will lead to cancellations.To make this idea precise, suppose that λ − λ ln ∈ (0, Λ n /r) and write where the function R n (λ) plays the role of an error term.Differentiating this identity with respect to λ, and noticing that f (λ) = − arccos λ, one immediately obtains that the bound where α 0 := min{0, −2s} , α 1 := min{1, −2s} .As usual, the constant in depends on δ.
By the mean value theorem, observing that R n (λ ln ) = 0, one then has from Equation (3.6) that with depending on δ.As the implicit constants are uniform in n and there are N ≈ r 1−γ intervals, this implies The leading contribution is therefore the constant in depending on δ.Note that the denominator is bounded from below because λ < 1 − δ.Thus, choosing γ ∈ (0, 1 2 ) and β sufficiently close to 1 (depending on γ and s), we conclude that |I 2 | r −η for some η > 0.
Let us now pass to the case s = 1 2 .Arguing as above, one can pick some β close to, but smaller than, 1 such that for some η > 0. For the sum going from l = 1 to r β , we can disregard the (1 − λ 2 l ) 1/2 term because The identity enables us to write The asymptotic expansions The quantity rO(λ 2 ) is of order r 2β −1 whenever l < r β .Fixing some β ∈ (0, 1 2 ), we therefore have Here we have used that Similarly, It only remains to consider the sum from r β to r β , where we can also assume that χ sm (λ l ) = 1.To this end, we define the function To show this sum goes to zero as r → ∞, we are going to exploit the cancellations of consecutive terms.For this, let us define By the mean value theorem, there exists some λ * ∈ (2kr −1 , (2k + 1)r −1 ) such that Let us finally consider the case s > 1 2 : Lemma 3.4.If s > 1  2 , there exists some η > 0 depending on s such that Here ζ is the Riemann's zeta function.
Proof.Let us use again the integral formula for Bessel functions to write e ir(sin x−sin y) e −i((l+m )x−(l+m)y) dx dy .
Applying the stationary phase argument [13, Theorem 7.7.5] with phase function sin x − sin y and amplitude e −i((l+m )x−(l+m)y) , one readily obtains the asymptotic expansion where the error term satisfies the pointwise bound Now, pick some β ∈ (0, 1  4 ) and write Then where the error term is bounded as This decay is smaller than r −1 if β < 1 4 .Expanding the sine, the above series can be computed in closed form in terms of the zeta function: To control the remaining term, we use that s > 1 2 and the bound for g λ proved in Lemma 3.1 to write As usual, the constant in depends on δ.The lemma then follows.
3.2.Intermediate frequency region.Our next goal is to derive bounds for the term e il(y−x) e ir(sin x−sin y) e i(my−m x) dx dy .
The difficulty here is that one cannot apply the standard stationary phase method as we did above because the critical points of the phase function ϕ l (x, y) := λ(y − x) + sin x − sin y are either degenerate or not uniformly non-degenerate.The main result is the following: Lemma 3.5.For any real s and all large enough r (depending on δ), |II| Cδ where C is independent of δ. Proof.Since when 1 − 2δ λ 1 + 2δ and δ 1, an elementary calculation shows that , where c is a positive constant that depends on δ.Therefore, take some χ(t) be a smooth nonnegative function that is equal to 1 for |t| < 100 δ 1/2 and 0 for |t| > 200 δ 1/2 .The non-stationary phase lemma then shows that e il(y−x) e ir(sin x−sin y) e i(my−m x) χ(x) χ(y) dx dy coincides with II modulo an exponentially small error.More precisely, for any N and some constant depending on N and δ.
To estimate II , let us start by defining z := y − x and writing e ilz e ir(sin(y−z)−sin y) e i((m−m )y+m z) χ(y − z) χ(y) dy dz .
A first step is to consider the sum and to relate it to its continuous counterpart Note that it is not a priori obvious that F (r, z) converges to S(r, z) as r → ∞ because, intuitively speaking, the sum is formally obtained by discretizing the integral with a "grid" of length 1/r, and r 1 is precisely the frequency at which the integrand oscillates.We proceed as follows.Firstly, write and note that e ilz r − The function h is smooth at the origin; in fact, h(z) = O(z).As moreover is bounded as with C a constant independent of z and δ.Since z will eventually be small, the fact that shows in which sense S(r, z) and F (r, z) are related.The reader can check that, had we argued as in (3.7), we would have obtained an error estimate of the form Cz, which is useless for our purposes.One can thus write e ir(sin(y−z)−sin y) e i((m−m )y+m z) R(r, z) χ(y − z)χ(y) 1 − h(z) dz dy The bound for R(r, z) and the fact that χ(t) is supported in |t| < 200 δ 1/2 immediately implies where the constant does not depend on δ.
To analyze II 1 , one cannot directly apply the stationary phase formula to the integral over R 3 because the critical set of the phase has dimension 1.Instead, let us define Then, the phase function ϕ y (λ, z) := λz + sin(y − z) has a unique critical point in the support of the integrand, (λ * , z * ) := (cos y, 0), and its Hessian is .
The stationary phase formula [13, Theorem 7.7.6]then ensures that, if r is large enough (depending on δ) with a constant independent of δ.Plugging this estimate into II 1 and using again that χ(t) is supported in |t| < 200 δ 1/2 , one finds with a constant independent of δ.Putting all the estimates together, the lemma is proven.
3.3.Large frequency region.The last lemma of this section shows that the contribution of the large frequencies is exponentially small: Lemma 3.6.For any N , |III| r −N for all large enough r (depending on δ).
Proof.Let us now use l as the large parameter in the formula for g λ l (r), which amounts to writing e il φλ l (x,y) e −i(m x−my) dx dy with φλ (x, y) := y − x + sin x − sin y λ .
where c δ is a positive constant that only depends on δ.Therefore, the non-stationary phase lemma [13, Theorem 7.7.1]ensures that g λ l (r) is an exponentially small function of l, meaning that for any N there exists a constant C (depending on δ and N ) such that This immediately implies that for any N , as claimed.
3.4.Asymptotics for series with derivatives of Bessel functions.The results we have derived above readily yield the asymptotic bounds for weighted sums of Bessel functions that we will crucially need in the next section.Specifically, Lemma 1.2 follows immediately by adding the estimates derived in the previous subsections and letting δ → 0 + .The explicit constants in the lemma are: One should observe that, to estimate the expected number of critical points of the random monochromatic wave (1.4), we will also need asymptotic information about series with derivatives of Bessel functions.This follows easily as a byproduct of Lemma 1.2 using the well-known recurrence relations In the following lengthly corollary of Lemma 1.2 we record the asymptotic formulas that we will need later on: Corollary 3.7.The following estimates hold:

Proof of Theorem 1.1
We are now ready to present the proof of the main theorem, which will consist of a number of steps.Recall that we defined the random function u as It will be apparent from the proof that the argument remains valid for much more general choices of σ l , for example of the form (1.6).Of course, the value of the constants κ(s), κ 3 2 , κ 5 2 one gets depends on the specific choice of σ l .

4.1.
A Kac-Rice formula.Our first objective is to derive an explicit, if hard to analyze, Kac-Rice type formula for the expected number of critical points of the Gaussian random function u.
In this subsection, we shall denote by the derivative and Hessian of u in polar coordinates.To apply the Kac-Rice expectation formula, let us start by showing that Du(r, θ) has a non-degenerate distribution: Lemma 4.1.The variance of the Gaussian random variable Du(r, θ) is recall the expression (4.1) for u(r, θ) and take advantage of the fact that u(r, θ) is real valued to write By the definition of the random variables a l , E(a l a l ) = 2δ l,l , so one obtains The same argument yields l|l| −2s J l (r) J l (r) = 0 by parity, and This easily implies that Var[Du(r, θ)] is a strictly positive matrix for all (r, θ).
Remark 4.2.The same computation as above shows that the covariance kernel of the random function (4.1) is The covariance kernel is therefore invariant under rotations but, in general, not under translation.An exception to this general fact is the case s = 0. Indeed, it is well known that the covariance kernel of ) by Graf's Addition Theorem.The corresponding spectral measure in this case is the Hausdorff measure on the unit circle.Observe that u will give the same asymptotics as u for s = 0 because, as we saw in Lemma 1.2, for s = 0 the series of Bessel functions is asymptotically of order 1 but the term J 0 (r) 2 decays like r −1 .By Lemma 4.3, their covariances Σ ij are then asymptotically equivalent.Note we have chosen to omit the term l = 0 in u for simplicity, especially when this term contributes to the asymptotic expansion (that is, for s > 1 2 in Lemma 1.2).Lemma 4.3.The expected value of the number of critical points of the random monochromatic wave (1.4) is where Proof.As Du(r, θ) is a non-degenerate Gaussian random variable by Lemma 4.1, the Kac-Rice integral formula in polar coordinates [3,Proposition 6.6] ensures that where ρ Du(r,θ) : R 2 → [0, ∞) denotes the probability distribution function of the R 2 -valued random variable Du(r, θ).
Next, let us reduce the computation of the conditional expectation to that of an ordinary expectation by introducing a new random variable ζ(r, θ).Just like D 2 u(r, θ), ζ(r, θ) will take values in the space of 2 × 2 symmetric matrices, which we shall henceforth identify with R 3 by labeling the matrix components of a symmetric matrix as where the linear operator B(r, θ) (which we can regard as a 3 × 2 matrix after identifying D 2 u(r, θ) with a 3-component vector) is chosen so that the covariance matrix of Du(r, θ) and ζ(r, θ) is 0: Indeed, one can plug (4.4) in the formula for E(ζ(r, θ) ⊗ Du(r, θ)) and check that E(ζ(r, θ) ⊗ Du(r, θ)) = 0 .As Du(r, θ) and ζ(r, θ) are jointly a Gaussian vector with zero mean, this condition ensures that they are independent random variables.This enables us to write the above conditional expectation as Arguing as in Lemma 4.1 and using that we have identified D 2 u(r, θ) with a 3-component vector, one finds that Var D 2 u(r) := E[D 2 u(r, θ) ⊗ D 2 u(r, θ)] is given by the 3 × 3 matrix Similarly, Combining these formulas, we derive that where Σ jk (r) are defined as in the statement of the lemma.
Let us now consider the Cholesky decomposition of this matrix: where the matrix M (r) is given by As the matrix Σ(r) is positive definite and ζ(r, θ) is a Gaussian random variable with zero mean and variance Σ(r), one then infers that the 3-component random variable is Gaussian, has zero mean and its variance matrix is the identity.It is thus straightforward that where is the probability density distribution of the random variable ζ(r, θ) and we have used the change of variables and the fact that the Jacobian determinant is det M (r) = (det Σ(r)) 1 2 .The lemma follows using that the probabability density function of the Gaussian random variable Du(r, θ) is as a consequence of the formula for Var Du(r, θ) computed in Lemma 4.1 and of the fact that the density function of an R k -valued Gaussian random variable Y with zero mean and variance matrix Σ is 4.2.Some technical lemmas.In the next subsections, we will discuss the behavior of the formula for the expected number of critical points that we have computed in Lemma 4.3 above.The analysis will strongly depend on the value of the parameter s.In the computations, we will use several technical lemmas repeatedly, often without further mention.
Proof.Consider any ε > 0 and assume, without any loss of generality, that o(q(r )) 0. By definition, there is some R ε such that o(q(r)) εq(r) for all r > R ε .Now set Q(r) := r r 0 q(r ) dr and write The following lemma will be very useful in the analysis of the asymptotic behavior of the number of critical points of u: The second term is obviously bounded as To estimate the first term, let As the function r a (log r) b is increasing for large enough r, we have if j is larger that a certain integer J a,b .With η = 0, 1, we can use the following asymptotic formula, which is an easy consequence of the Euler-Maclaurin formula, to derive the formula of the statement.Here we have used that πJ = R + O(1) and that the integral over r ∈ [π, πJ a,b ] is obviously bounded independently of R.
Before discussing the behavior of EN (∇u, R) in the different regularity regimes, one should note that the integral appearing in Lemma 4.3 is remarkably hard to analyze.We will be able to obtain much more convenient integral representations by means of the following lemma: Lemma 4.7.Let A, B, C be real constants.Then where Proof.Defining the matrix one can write the above integral as The results about Gaussian integrals involving an absolute value function derived in [15, Theorem 2.1] therefore ensure that Now a straightforward computation yields the formula in the statement.
4.3.The case s < 1 2 .We are ready to compute the asymptotics for the number of critical points when s < 1 2 : Proof.Let us compute the matrix Σ(r).From Equation (4.7) and the asymptotic formulas for sums of Bessel functions recorded in Corollary 3.7, it follows that where the leading contribution is and the error is bounded as Here and in what follows, o(1) denotes a quantity that tends to zero as r → ∞.
Let us define and note that, by the formula for Σ(r) and the asymptotics for weighted sums of Bessel functions presented in Corollary 3.7, Likewise, the quantity σ(r) := Σ 11 (r) Σ 22 (r) (4.11) satisfies the asymptotic bound Finally, the integral can be then estimated, as a consequence of Lemmas 4.4 and 4.7 and of the preceding asymptotic bounds, as where κ(s) is defined as in the statement.Thus, the integral formula in Lemma 4.3 ensures that In the next lemma, we analyze the behavior of the positive constant κ(s) (which is written simply as κ(s) in the statement of Theorem 1.1), for s < 1  2 .The key idea is to obtain an easier characterization of this constant as a one-dimensional integral.Interestingly, the global maximum of κ(s) is attained at s = 0, that is, in the classical case of random waves with a translation-invariant covariance kernel.In Figure 2 we have plotted κ(s) for the first region of s < 1/2 using the next lemma.Lemma 4.9.The function κ(s) is smooth, strictly increasing on s ∈ (−∞, 0), and strictly decreasing on (0, 1 2 ).Furthermore, Proof.The limiting values can be computed directly from the formula for κ(s).Indeed, the (somewhat surprising) fact that κ(s) → 0 as s → −∞ is obvious in view of Equation (4.9), and as is the limit lim To analyze the behavior of κ(s) for intermediate values of s, we use Lemma 4.7 to rewrite (4.9) as

Note that
Using that the polynomials appearing on the numerators are all positive for t > 0 and s < 1  2 , it follows that κ (s)/s < 0 for all s ∈ (−∞, 0) ∪ (0, 1  2 ).The result then follows.Remark 4.10.In the case s = 0, where κ(s) attains its maximum, we recover the well-known asymptotic formula (see Appendix B) for the expected number of critical points: where we have used that for s = 0 the integral above becomes 4.4.The case s = 1 2 .We shall next show that, in spite of the appearance of logarithmic terms in the formulas, the asymptotic behavior in the case s = 1 2 coincides with the limit as s → 1 2 + of the formula derived in Lemma 4.8.
Proof.From Equation (4.7) and Corollary 3.7, we infer that in the case s = 1 2 , we can write where and the error is bounded as Likewise, the function σ(r) defined in (4.11) satisfies Plugging these formulas in (4.12), we obtain 4.5.The case 1 2 < s < 3 2 .We shall next show that, in the regime 1 2 < s < 3 2 , the expected number of critical points contained in a large disk also grows like the area.The associated proportionality constant, which we denote by κ(s), turns out to be smooth on (−∞, 1  2 )∪( 1 2 , 3 2 ) but only continuous at s = 1 2 .
Similarly, and using the same notation as in the last two subsections, One can then plug these formulas in (4.12) to find As 2 1−2s < 1, this immediately implies dr .
for all |b| < 1, the formula of the statement now follows using Lemma 4.6.
Remark 4.13.It follows from Lemmas 4.8, 4.11 and 4.12 that κ(s ]), and that κ(s) is Lipschitz at s = 1 2 but not C 1 .It also follows that 4.6.The case s = 3 2 .Here we shall see that the expected number of critical points contained in a ball of large radius does not grow like the area of the ball any longer: Proof.The argument is essentially as before.Using Corollary 3.7 and Equation (4.7), one can write Σ(r) = σ 0 (r) + R(r), with Hence, keeping track of the errors using Lemmas 4.4-4.5 as before, This readily implies , so Lemma 4.3 ensures that the expected number of critical points satisfies The asymptotic behavior of this integral is by Lemma 4.6, so the result follows.
4.8.The case s = 5 2 .The next lemma shows that at this regularity level, there is another transition in the asymptotic behavior of the expected number of critical points of u: 4.9.The case s > 5  2 .In this regime, the proof goes as before, showing that the expected number of critical points contained in a large ball grows asymptotically like the radius.However, the explicit formulas one obtains for the proportionality constant are extremely cumbersome.
Lemma 4.17.For s > 5  2 , there exists an explicit constant κ(s) > 0 such that Note that all the nonzero matrix components are exactly of order 1/r.While this fact does not make the problem any harder from a conceptual point of view, it leads to cumbersome expressions for the various quantities appearing in the equations.Specifically, it is not hard to show that Plugging this formula in the expression for I(r, z), one finds that where the constants are smooth functions of sin 2r.Lemma 4.7 then shows that for some explicit smooth function of the form  One can now read the asymptotic behavior of EN (∇u, R) in any regularity regime from the lemmas that we have established in this section.Theorem 1.1 is therefore proven.

Asymptotics for the number of critical points in the high regularity case
This section is devoted to the proof of Theorem 1.3.As all along this paper, we shall take the definition (4.1) for the Gaussian random function u.
5.1.Some non-probabilistic lemmas.Before presenting the proof of this theorem, we need to prove a few auxiliary results that do not use the fact that u and f are random functions.Specifically, these lemmas concern solutions to the Helmholtz equation on R 2 of the form v(x) := T e −ix•E(φ) g(φ) dφ where g ∈ H m (T) for a certain real m and the standard embedding E : T → R 2 is given by (1.2).
We start by recalling the following result on the asymptotic behavior of v, which we proved in [10,Proposition 2.2 and Remark 3.2].In what follows, we will denote the real and imaginary parts of a function g by g R and g I , respectively.g I (θ) sin(r where the errors are bounded as The following theorem provides very precise asymptotic information about the critical points of v: Lemma 5.2.Assume that m > 9/2, that g does not vanish on T, and that all the critical points of |g| are non-degenerate.If φ * is a critical point of |g|, then for each large enough positive integer n there exists a critical point (r * n , θ * n ) of v such that Conversely, if (r * , θ * ) is a critical point of v, there is some critical point φ * of |g| such that Proof.Let us consider the function ) , whose critical points (r * , θ * ) are the solutions to the equations Im g(θ * )e −i(r * − π 4 ) = 0 , Re g (θ * )e −i(r * − π 4 ) = 0 .
Writing g = |g|e i arg g , an elementary calculation shows that (r * , θ * ) is a critical point of V if and only if r * = arg g(θ * ) + π 4 + πn for some integer n and Re[g(θ * )g (θ * ) = 0.As g does not vanish on T, the latter condition simply means that θ * is a critical point of |g|.Furthermore, the Hessian of V at the critical points is As the critical points of V are uniformly non-degenerate by (5.1), Thom's isotopy theorem (as stated, e.g., in [8]) ensures that v has a critical point at a distance at most C/n to each of the critical points (r * , θ * ) of V as described above, provided that n is large enough.Furthermore, the asymptotic formulas for Dv presented in Lemma 5.1 guarantee that all critical points of v that are far enough from the origin must be of this form.The lemma is then proven. .Therefore, if one can prove that, with probability 1, f does not vanish on T and all the critical points of |f | are nondegenerate, Theorem 1.3 will follow as an easy consequence of Lemma 5.2.
Proving the first part of this assertion is completely standard, but the second part is quite harder.In both cases, the proof relies on Bulinskaya's lemma, which one can state as follows [3, Proposition 6.11]: Lemma 5.3 (Bulinskaya).Let Y : T → R 2 be a random function that is of class C 1 (T) almost surely.For each φ ∈ T, assume that the random variable Y (φ) has a probability density ρ Y (φ) : R 2 → [0, ∞) that is bounded in some fixed neighborhood of the origin.Then P{Y (φ) = 0 for some φ ∈ T} = 0 .
Armed with Bulinskaya's lemma, it is easy to show that, almost surely, f does not vanish: Lemma 5.4.With probability 1, f does not vanish on T.
Proof.By the definition of u, cf.Equations (4.1) and (1.5), Ỹ (φ) := (f R (φ), f I (φ)) is a Gaussian random field Ỹ : T → R 2 with zero mean.The covariance of Ỹ (φ) can be computed just as in Lemma 4.1, obtaining the nondegenerate matrix Therefore, Ỹ (φ) has a bounded probability density function on R 2 because Σ is a nondegenerate matrix.Lemma 5.3 then ensures that Ỹ does not vanish with probability 1.As the zeros of Ỹ and f obviously coincide, the lemma follows.
The crux of the proof of Theorem 1.3 is to show that the critical points of |f | are nondegenerate.This is not direct because |f | is not a Gaussian variable, and showing that it has a bounded probability density requires some work.The main ingredient of the proof is the estimate we present in the following lemma.The proof is somewhat involved, so we have relegated it to the next subsection in order to streamline the presentation of the proof of Theorem 1.3.To state the auxiliary result, we will write points in R 6 as with z := (z 1 , z 2 , z 3 , z 4 ) and z := (z 5 , z 6 ).Lemma 5.5.Consider the nonnegative rational function on R 6 given by (5.2) For any constant c > 0, Assuming for the moment that this technical lemma holds, proving that the critical points of |f | are nondegenerate almost surely is straightforward: Lemma 5.6.With probability 1, all the critical points of |f | are nondegenerate.
Proof.Let us start by noting that Therefore, all the critical points of |f | are nondegenerate if and only if As Y ∈ C 2 (T) almost surely because s > 5, in order to apply Bulinskaya's lemma we only need to show that Y (φ) has a probability density that is bounded in a neighborhood of the origin.The random variable Y (φ) is obviously not Gaussian, so in order to compute its density we need to argue in an indirect way.
The starting point is the fact that the 2-jet of f , which does not depend on φ, can be computed from the definition by arguing just as in the proof of Lemma 4.1.It turns out that Var Z(φ) = Σ, where Σ is the 6 × 6 matrix , where and we have set σ l := |l| −s for l = 0 and σ 0 := 0. We have chosen to write this formula in terms of σ l so that it is apparent that the result only uses the asymptotic properties of the sequence σ l .Note that these sums are all convergent because s > 5.
The determinant of Σ is i strictly by the Cauchy-Schwartz inequality, the matrix Σ is invertible.Therefore, for each φ ∈ T, the probability density distribution of Z(φ) is given by the Gaussian function Consider now the map H : R 6 → R 6 given by This map is invertible outside the hyperplane {z 2 = 0}, with inverse and its corresponding Jacobian determinant is det ∇H −1 (z) = −z −2 2 .Therefore, the probability density distribution of the random variable H[Z(φ)] is obtained by pulling back with the map H the probability distribution of Z(φ): . Now let H : R 6 → R 2 denote the last two components of the map (5.3), that is, As the random variables Y (φ) and Z(φ) are related by it then follows from (5.4) that the density of Y (φ) is given by the marginal distribution ρ H[Z(φ)] (z) dz .Now notice that the function Q(z) defined in (5.2) is simply As the matrix Σ is positive definite, therefore there is a positive constant c > 0 such that ρ Y (φ) (z ) The map z → ( , τ, z 3 , z 4 ) is invertible outside the hyperplane z 3 = 0 and the set τ = 0.In terms of these variables, the integral reads as (5.5) As Q 1 is a second order polynomial in z 4 , one can explicitly integrate in this variable, obtaining Let us now consider polar coordinates (σ, α) ∈ R + × T, defined as z 3 =: σ cos α , τ =: σ sin α .
Still denoting by Q 2 the expression of (5.5) in these variables, and similarly with the other functions Q j , we get We can then write Here we have used that for some C > 0 and for all a ∈ R. To integrate in , we need that Thus, we obtain the bound I 2 1 , from the fact that the logarithmic singularities at σ = 0 and σ = 1/ √ 2 are integrable.Lemma 5.5 in then proven.

Theorem 1 . 1 . 2 , κ 5 2
For any real s, the following statements hold: (i) There exist explicit positive constants κ(s), κ 3 such that the expected number of critical points of the Gaussian random function u satisfies

3 2−Figure 1 .
Figure 1.Consider the asymptotic behavior of EN (∇u, R) ∼ κ(s)R e(s) proved in Theorem 1.1.In red, we have plotted the exponent e(s) as a function of s ∈ R\{ 3 2 , 5 2 }.Logarithmic effects appear at the endpoints s = 3/2 and s = 5/2.In blue, we have plotted κ(s) in the region where the asymptotic growth is volumetric, s < 1 2 .The maximum of κ(s) in this region is attained at s = 0 and that κ(s) is not continuously differentiable at s = 1/2.The reader can find a plot of κ(s) in the range s ∈ ( 3 2 , 5 2 ) in Figure 3, cf.Section 4. Note that κ(s) = EN (|f | )/π by Theorem 1.3.

Lemma 4 . 6 . 0 P
Consider a positive smooth π-periodic function P and constants a 0 and b ∈ R. If a = 0, we also assume that b 0.Then, for R 1, R π r a (log r) b P (r) dr ∼ R a+1 (log R) b π(a + 1) π (r) dr .Proof.Let us define J := R/π and write R = Jπ + R 1 , with 0 R 1 < π.We can then write R π r a (log r) b P (r) dr = r a (log r) b P (r) dr + πJ+R 1 πJ r a (log r) b P (r) dr .

2 ρ 5 . 3 .
ensures that sup |z |< 1 Y (φ) (z ) 1. Lemma 5.3 then guarantees that the random function Y does not vanish on T almost surely, and the theorem follows.Theorem 1.3 is then proven, modulo the proof of Lemma 5.5, which we will address next.Proof of the main technical lemma.Let us now present the proof of Lemma 5.5.To make the exposition clearer, we will divide the proof in three steps.5.3.1.The integral I.The first step is to rewrite the integral convenient way.For this, let us set := z 1 z 3 − z 5 , τ := z 1 z 3 − z 5 z 2 .