arXiv subjects

Alejandro Rivera

Publications and source records attributed to Alejandro Rivera.

13 recordsLinked to original sources

arXiv2021

Existence of an unbounded nodal hypersurface for smooth Gaussian fields in dimension $d \ge 3$

For the Bargmann--Fock field on $\mathbb R^d$ with $d\ge3$, we prove that the critical level $\ell_c(d)$ of the percolation model formed by the excursion sets $\{ f \ge \ell \}$ is strictly positive. This implies that for every $\ell$ sufficiently close to $0$ (in particular for the nodal hypersurfaces corresponding to the case $\ell=0$), $\{f=\ell\}$ contains an unbounded connected component that visits "most" of the ambient space. Our findings actually hold for a more general class of positively correlated smooth Gaussian fields with rapid decay of correlations. The results of this paper show that the behaviour of nodal hypersurfaces of these Gaussian fields in $\mathbb R^d$ for $d\ge3$ is very different from the behaviour of nodal lines of their two-dimensional analogues.

math.PR↗

arXiv2021

On the limiting behaviour of arithmetic toral eigenfunctions

We consider a wide class of families $(F_m)_{m\in\mathbb{N}}$ of Gaussian fields on $\mathbb{T}^d=\mathbb{R}^d/\mathbb{Z}^d$ defined by \[F_m:x\mapsto \frac{1}{\sqrt{|\Lambda_m|}}\sum_{\lambda\in\Lambda_m}\zeta_\lambda e^{2\pi i\langle \lambda,x\rangle}\] where the $\zeta_\lambda$'s are independent std. normals and $\Lambda_m$ is the set of solutions $\lambda\in\mathbb{Z}^d$ to $p(\lambda)=m$ for a fixed elliptic polynomial $p$ with integer coefficients. The case $p(x)=x_1^2+\dots+x_d^2$ is a random Laplace eigenfunction whose law is sometimes called the $\textit{arithmetic random wave}$, studied in the past by many authors. In contrast, we consider three classes of polynomials $p$: a certain family of positive definite quadratic forms in two variables, all positive definite quadratic forms in three variables except multiples of $x_1^2+x_2^2+x_3^2$, and a wide family of polynomials in many variables. For these classes of polynomials, we study the $(d-1)$-dimensional volume $\mathcal{V}_m$ of the zero set of $F_m$. We compute the asymptotics, as $m\to+\infty$ along certain sequences of integers, of the expectation and variance of $\mathcal{V}_m$. Moreover, we prove that in the same limit, $\frac{\mathcal{V}_m-\mathbb{E}[\mathcal{V}_m]}{\sqrt{\text{Var}(\mathcal{V}_m)}}$ converges to a std. normal. As in previous works, one reduces the problem of these asymptotics to the study of certain arithmetic properties of the sets of solutions to $p(\lambda)=m$. We need to study the number of such solutions for fixed $m$, the number of quadruples of solutions $(\lambda,\mu,\nu,\iota)$ satisfying $\lambda+\mu+\nu+\iota=0$, ($4$-correlations), and the rate of convergence of the counting measure of $\Lambda_m$ towards a certain limiting measure on the hypersurface $\{p(x)=1\}$. To this end, we use prior results on this topic but also prove a new estimate on correlations, of independent interest.

math.PR↗

arXiv2020

How Lagrangian states evolve into random waves

In this paper, we consider a compact manifold $(X,d)$ of negative curvature, and a family of semiclassical Lagrangian states $f_h(x) = a(x) e^{\frac{i}{h} \phi(x)}$ on $X$. For a wide family of phases $\phi$, we show that $f_h$, when evolved by the semiclassical Schr\"odinger equation during a long time, resembles a random Gaussian field. This can be seen as an analogue of Berry's random waves conjecture for Lagrangian states.

math-ph↗

arXiv2020

The phase transition for planar Gaussian percolation models without FKG

We develop techniques to study the phase transition for planar Gaussian percolation models that are not (necessarily) positively correlated. These models lack the property of positive associations (also known as the `FKG inequality'), and hence many classical arguments in percolation theory do not apply. More precisely, we consider a smooth stationary centred planar Gaussian field $f$ and, given a level $\ell \in \mathbb{R}$, we study the connectivity properties of the excursion set $\{f \geq -\ell\}$. We prove the existence of a phase transition at the critical level $\ell_{crit}=0$ under only symmetry and (very mild) correlation decay assumptions, which are satisfied by the random plane wave for instance. As a consequence, all non-zero level lines are bounded almost surely, although our result does not settle the boundedness of zero level lines (`no percolation at criticality'). To show our main result: (i) we prove a general sharp threshold criterion, inspired by works of Chatterjee, that states that `sharp thresholds are equivalent to the delocalisation of the threshold location'; (ii) we prove threshold delocalisation for crossing events at large scales -- at this step we obtain a sharp threshold result but without being able to locate the threshold -- and (iii) to identify the threshold, we adapt Tassion's RSW theory replacing the FKG inequality by a sprinkling procedure. Although some arguments are specific to the Gaussian setting, many steps are very general and we hope that our techniques may be adapted to analyse other models without FKG.

math.PR↗

arXiv2019

Talagrand's inequality in planar Gaussian field percolation

Let f be a stationary isotropic non-degenerate Gaussian field on R^2. Assume that f = q * W where q is both C^2 and L^2 and W is the L^2 white noise on R^2. We extend a result by Stephen Muirhead and Hugo Vanneuville by showing that, assuming that q * q is pointwise non-negative and has fast enough decay, the set {f > -l} percolates with probability one when l > 0 and with probability zero if l < 0 or l = 0. We also prove exponential decay of crossing probabilities and uniqueness of the unbounded cluster. To this end, we study a Gaussian field g defined on the torus and establish a superconcentration formula for the threshold T(g) which is the minimal value such that {g > -T(g)} contains a non-contractible loop. This formula follows from a Gaussian Talagrand type inequality.

math.PR↗

arXiv2018

A covariance formula for topological events of smooth Gaussian fields

We derive a covariance formula for the class of `topological events' of smooth Gaussian fields on manifolds; these are events that depend only on the topology of the level sets of the field, for example (i) crossing events for level or excursion sets, (ii) events measurable with respect to the number of connected components of level or excursion sets of a given diffeomorphism class, and (iii) persistence events. As an application of the covariance formula, we derive strong mixing bounds for topological events, as well as lower concentration inequalities for additive topological functionals (e.g. the number of connected components) of the level sets that satisfy a law of large numbers. The covariance formula also gives an alternate justification of the Harris criterion, which conjecturally describes the boundary of the percolation university class for level sets of stationary Gaussian fields. Our work is inspired by a recent paper by Rivera and Vanneuville, in which a correlation inequality was derived for certain topological events on the plane, as well as by an old result of Piterbarg, in which a similar covariance formula was established for finite-dimensional Gaussian vectors.

math.PR↗

arXiv2018

A lower bound for the Bogomolny-Schmit constant for random monochromatic plane waves

This note deals with nodal domains of random monochromatic plane waves. It was shown by Nazarov and Sodin that the expected number of such nodal domains included in a disk of radius $R$ is proportional to $\pi R^2$ in the large $R$ limit. However, very little is known on the value of the proportionality constant from a mathematical point of view. The aim of this note is to obtain a lower bound on the value of this constant my elementary means.

math-ph↗

arXiv2018

Weighted local Weyl laws for elliptic operators

Let $A$ be an elliptic pseudo-differential operator of order $m$ on a closed manifold $\mathcal{X}$ of dimension $n>0$, formally positive self-adjoint with respect to some positive smooth density $d\mu_\mathcal{X}$. Then, the spectrum of $A$ is made up of a sequence of eigenvalues $(\lambda_k)_{k\geq 1}$ whose corresponding eigenfunctions $(e_k)_{k\geq 1}$ are $C^\infty$ smooth. Fix $s\in\mathbb{R}$ and define \[ K_L^s(x,y)=\sum_{0<\lambda_k\leq L}\lambda_k^{-s} e_k(x)\overline{e_k(y)}\, .\] We derive asymptotic formulae near the diagonal for the kernels $K_L^s(x,y)$ when $L\rightarrow +\infty$ with fixed $s$. For $s=0$, $K^0_L$ is the kernel of the spectral projector studied by H\"ormander in \cite{ho68}. In the present work we build on H\"ormander's result to study the kernels $K^s_L$. If $s<\frac{n}{m}$, $K_L^s$ is of order $L^{-s+n/m}$ and near the diagonal, the rescaled leading term behaves like the Fourier transform of an explicit function of the symbol of $A$. If $s=\frac{n}{m}$, under some explicit generic condition on the principal symbol of $A$, which holds if $A$ is a differential operator, the kernel has order $\ln(L)$ and the leading term has a logarithmic divergence smoothed at scale $L^{-1/m}$. Our results also hold for elliptic differential Dirichlet eigenvalue problems.

math.SP↗

arXiv2018

Expected number of nodal components for cut-off fractional Gaussian fields

Let $({\mathcal{X}},g)$ be a closed Riemmanian manifold of dimension $n>0$. Let $\Delta$ be the Laplacian on ${\mathcal{X}}$, and let $(e\_k)\_k$ be an $L^2$-orthonormal and dense family of Laplace eigenfunctions with respective eigenvalues $(\lambda\_k)\_k$. We assume that $(\lambda\_k)\_k$ is non-decreasing and that the $e\_k$ are real-valued. Let $(\xi\_k)\_k$ be a sequence of iid $\mathcal{N}(0,1)$ random variables. For each $L>0$ and $s\in{\mathbb{R}}$, possibly negative, set\[f^s\_L=\sum\_{0<\lambda\_j\leq L}\lambda\_j^{-\frac{s}{2}}\xi\_je\_j\, .\]Then, $f\_L^s$ is almost surely regular on its zero set. Let $N\_L$ be the number of connected components of its zero set. If $s<\frac{n}{2}$, then we deduce that there exists $\nu=\nu(n,s)>0$ such that $N\_L\sim \nu {Vol}\_g({\mathcal{X}})L^{n/2}$ in $L^1$ and almost surely. In particular, ${\mathbb{E}}[N\_L]\asymp L^{n/2}$. On the other hand, we prove that if $s=\frac{n}{2}$ then\[{\mathbb{E}}[N\_L]\asymp \frac{L^{n/2}}{\sqrt{\ln\left(L^{1/2}\right)}}\, .\]In the latter case, we also obtain an upper bound for the expected Euler characteristic of the zero set of $f\_L^s$ and for its Betti numbers. In the case $s>n/2$, the pointwise variance of $f\_L^s$ converges so it is not expected to have universal behavior as $L\rightarrow+\infty$.

math.PR↗

arXiv2017

Quasi-independence for nodal lines

We prove a quasi-independence result for level sets of a planar centered stationary Gaussian field with covariance $(x,y)\mapsto\kappa(x-y)$. As a first application, we study percolation for nodal lines in the spirit of [BG16]. In the said article, Beffara and Gayet rely on Tassion's method ([Tas16]) to prove that, under some assumptions on $\kappa$, most notably that $\kappa \geq 0$ and $\kappa(x)=O(|x|^{-325})$, the nodal set satisfies a box-crossing property. The decay exponent was then lowered to $16+\varepsilon$ by Beliaev and Muirhead in [BM17]. In the present work we lower this exponent to $4+\varepsilon$ thanks to a new approach towards quasi-independence for crossing events. This approach does not rely on quantitative discretization. Our quasi-independence result also applies to events counting nodal components and we obtain a lower concentration result for the density of nodal components around the Nazarov and Sodin constant from [NS15].

math.PR↗

arXiv2017

The critical threshold for Bargmann-Fock percolation

In this article, we study the excursions sets $\mathcal{D}\_p=f^{-1}([-p,+\infty[)$ where $f$ is a natural real-analytic planar Gaussian field called the Bargmann-Fock field. More precisely, $f$ is the centered Gaussian field on $\mathbb{R}^2$ with covariance $(x,y) \mapsto \exp(-\frac{1}{2}|x-y|^2)$. In [BG16], Beffara and Gayet prove that, if $p \leq 0$, then a.s. $\mathcal{D}\_p$ has no unbounded component. We show that conversely, if $p>0$, then a.s. $\mathcal{D}\_p$ has a unique unbounded component. As a result, the critical level of this percolation model is $0$. We also prove exponential decay of crossing probabilities under the critical level. To show these results, we develop several tools including a KKL-type result for biased Gaussian vectors (based on the analogous result for product Gaussian vectors by Keller, Mossel and Sen in [KMS12]) and a sprinkling inspired discretization procedure. These intermediate results hold for more general Gaussian fields, for which we prove a discrete version of our main result.

math.PR↗

arXiv2016

Anomalies in local Weyl laws and applications to random topology at critical dimension

Let $\mathcal{M}$ be a smooth manifold of positive dimension $n$ equipped with a smooth density $d\mu_{\mathcal{M}}$. Let $A$ be a polyhomogeneous elliptic pseudo-differential operator of positive order $m$ on $\mathcal{M}$ which is symmetric for the $L^2$ scalar product defined by $d\mu_{\mathcal{M}}$. For each $L>0$, the space $U_L=\bigoplus_{\lambda\leq L}Ker(A-\lambda Id)$ is a finite dimensional subspace of $C^\infty(\mathcal{M})$. Let $\Pi_L$ be the spectral projector onto $U_L$. Given $s\in\mathbb{R}$, we compute the asymptotics of the integral kernel $K_L$ of $\Pi_LA^{-s}$ in the cases where $n>ms$ and $n=ms$ respectively. Next, assuming that $\mathcal{M}$ is closed, let $(e_n)_{n\in\mathbb{N}}$ and $(\lambda_n)_{n\in\mathbb{N}}$ be the sequence of $L^2$ normalized eigenfunctions and eigenvalues of $A$ where the latter sequence organized in increasing order. Let $(\xi_n)_{n\in\mathbb{N}}$ be a sequence of independent centered gaussians of variance $1$. We fix a parameter $s\in\mathbb{R}$ such that $n\geq ms$ and consider the family $(\phi_L)_{L>0}$ of smooth random fields on $\mathcal{M}$ defined by \[\phi_L=\sum_{0<\lambda_j\leq L}\lambda_j^{-\frac{s}{2}}\xi_je_j\] for each $L>0$. It turns out that the covariance function of $\phi_L$ is $K_L$. Using this information, we apply the derived asymptotics to study the zero set of $\phi_L$. If $n>ms$ then the number of components of the zero set of $\phi_L$ concentrates around $aL^{\frac{n}{m}}$ for some positive constant $a$. On the other hand, if $n=ms$, each Betti number of the zero set has an expectation bounded by $C\ln\Big(L^{\frac{1}{m}}\Big)^{-\frac{1}{2}}L^{\frac{n}{m}}$ where $C$ is an explicit constant. When $\mathcal{M}$ is a closed surface with a Riemmanian metric, $A$ is the Laplacian and $d\mu_\mathcal{M}$ is the Riemmanian volume, $C$ equals $\frac{1}{4\pi^2}\sqrt{\frac{3}{2}}Vol(\mathcal{M})$.

math.SP↗

arXiv2016

Hole probability for nodal sets of the cut-off Gaussian Free Field

Let ($\Sigma$, g) be a closed connected surface equipped with a riemannian metric. Let ($\lambda$ n) n$\in$N and ($\psi$ n) n$\in$N be the increasing sequence of eigenvalues and the sequence of corresponding L 2-normalized eigenfunctions of the laplacian on $\Sigma$. For each L \textgreater{} 0, we consider $\phi$ L = 0\textless{}$\lambda$n$\le$L $\xi$n $\sqrt$ $\lambda$n $\psi$ n where the $\xi$ n are i.i.d centered gaussians with variance 1. As L $\rightarrow$ $\infty$, $\phi$ L converges a.s. to the Gaussian Free Field on $\Sigma$ in the sense of distributions. We first compute the asymptotic behavior of the covariance function for this family of fields as L $\rightarrow$ $\infty$. We then use this result to obtain the asymptotics of the probability that $\phi$ L is positive on a given open proper subset with smooth boundary. In doing so, we also prove the concentration of the supremum of $\phi$ L around 1 $\sqrt$ 2$\pi$ ln L.

math.PR↗