arXiv subjects

Pierre Yves Gaudreau Lamarre

Publications and source records attributed to Pierre Yves Gaudreau Lamarre.

At least 19 recordsLinked to original sources

arXiv2026

The Parabolic Anderson Model's Total Mass at Small Times: Geometry, Fluctuations, and Renormalization

Let $D\subset\mathbb R^d$ be a bounded domain, let $κ>0$ be fixed, and let $W$ be a fractional Brownian sheet on $\mathbb R\times\mathbb R^d$. Consider the Stratonovich parabolic Anderson model (PAM) $\partial_tu_κ=(\frac12Δ+κW')u_κ$ with Dirichlet boundary condition on $D$ and the flat initial condition $u_κ(0,\cdot)=\mathbf 1_D$. We calculate exact asymptotics for the expectation and the standard deviation of the total mass $\int_Du_κ(t,x)~\mathrm d x$ as $t\to0$ under the assumption that $W$'s Hurst indices are all at least $1/2$ and that $u_κ$'s moments are finite for small enough $t>0$. In doing so, we uncover that these asymptotics are determined by a competition between three mechanisms: (1) $\mathbf{Geometry}$: The rate of heat diffusion through the boundary $\partial D$. (2) $\mathbf{Fluctuations}$: $W$'s time Hurst index. (3) $\mathbf{Renormalization}$: The singularity of deterministic Stratonovich corrections. As a result, we identify novel phase transition phenomena, which arise from the influence of $W$'s Hurst indices on the relative magnitudes of these contributions.

math.PR↗

arXiv2026

Optimal Covariance Estimates for Schrödinger Semigroups with White Noise in $d=1,2$

For $d\in\{1,2\}$, let $H=-\frac{1}{2}Δ+ V +ξ$ be the random Schrödinger operator on $L^2(\mathbb{R}^d)$ where $ξ$ is a standard Gaussian white noise and $V$ is a deterministic potential with power-law growth at infinity. Using a Feynman-Kac formula for the trace of the Schrödinger semigroup, we give optimal asymptotic upper and lower bounds on the covariance of $\mathrm{Tr}[e^{-sH}]$ and $\mathrm{Tr}[e^{-tH}]$ as $s,t\to0$ through estimates on Brownian bridge local times. These estimates are a significant improvement on previous bounds in the case $d=1$ and are the first of their kind for $d=2$. As an application of these new estimates, we prove a quantitative hyperuniformity-type property and decorrelation rate for the trace as $s,t\to0$.

math.PR↗

arXiv2026

On the Spectral Geometry and Small Time Mass of Anderson Models on Planar Domains

We consider the Anderson Hamiltonian (AH) and the parabolic Anderson model (PAM) with white noise and Dirichlet boundary condition on a bounded planar domain $D\subset\mathbb R^2$. We compute the small time asymptotics of the AH's exponential trace up to order $O(\log t)$, and of the PAM's mass up to order $O(t\log t)$. Our proof is probabilistic, and relies on the asymptotics of intersection local times of Brownian motions and bridges in $\mathbb R^2$. Applications of our main result include the following: (i) If the boundary $\partial D$ is sufficiently regular, then $D$'s area and $\partial D$'s length can both be recovered almost surely from a single observation of the AH's eigenvalues. This extends Mouzard's Weyl law in the special case of bounded domains (Ann. Inst. H. Poincaré Probab. Statist. 58(3): 1385-1425). (ii) If $D$ is simply connected and $\partial D$ is fractal, then $\partial D$'s Minkowski dimension (if it exists) can be recovered almost surely from the PAM's small time asymptotics. (iii) The variance of the white noise can be recovered almost surely from a single observation of the AH's eigenvalues.

math.PR↗

arXiv2025

Recovering Parameters from Edge Fluctuations: Beta-Ensembles and Critically-Spiked Models

Let $Λ=\{Λ_0,Λ_1,Λ_2,\ldots\}$ be the point process that describes the edge scaling limit of either (i) "regular" beta-ensembles with inverse temperature $β>0$, or (ii) the top eigenvalues of Wishart or Gaussian invariant random matrices perturbed by $r_0\geq1$ critical spikes. In other words, $Λ$ is the eigenvalue point process of one of the scalar or multivariate stochastic Airy operators. We prove that a single observation of $Λ$ suffices to recover (almost surely) either (i) $β$ in the case of beta-ensembles, or (ii) $r_0$ in the case of critically-spiked models. Our proof relies on the recently-developed semigroup theory for the multivariate stochastic Airy operators. Going beyond these parameter-recovery applications, our results also (iii) refine our understanding of the rigidity properties of $Λ$, and (iv) shed new light on the equality (in distribution) of stochastic Airy spectra with different dimensions and the same Robin boundary conditions.

math.PR↗

arXiv2025

On the Rigidity of Projected Perturbed Lattices

We study the occurrence of number rigidity and deletion singularity in a class of point processes that we call {\it projected perturbed lattices}. These are generalizations of processes of the form $Π=\{\|z\|^α+g_z\}_{z\in\mathbb{Z}^d}$ where $(g_z)_{z\in\mathbb{Z}^d}$ are jointly Gaussian, $α>0$, $d\in\mathbb{N}$, and $\|\cdot\|$ is a norm. We develop a new technique to prove sufficient conditions for the deletion singularity of $Π$, which improves significantly on the conditions one can obtain using the standard rigidity toolkit (e.g., the variance of linear statistics). In particular, we obtain the first lower bounds on $α$ for the deletion singularity of $Π$ that are independent of the dimension $d$ and the correlation of the $g_z$'s.

math.PR↗

arXiv2025

Trace Moments for Schrödinger Operators with Matrix White Noise and the Rigidity of the Multivariate Stochastic Airy Operator

We study the semigroups of random Schrödinger operators of the form $\widehat{H}f=-\frac12f''+(V+ξ)f$, where $f:I\to\mathbb F^r$ ($\mathbb F=\mathbb R,\mathbb C,\mathbb H$) are vector-valued functions on a possibly infinite interval $I\subset\mathbb R$ that satisfy a mix of Robin and Dirichlet boundary conditions, $V$ is a deterministic diagonal potential with power-law growth at infinity, and $ξ$ is a matrix white noise. Our main result consists of Feynman-Kac formulas for trace moments of the form $\mathbf E[\prod_{k=1}^n\mathrm{Tr}[\mathrm e^{-t_k\widehat{H}}]]$ ($n\in\mathbb N$, $t_k>0$). One notable example covered by our main result consists of the multivariate stochastic Airy operator (SAO) of Bloemendal and Virág (Ann. Probab., 44(4):2726-2769, 2016), which characterizes the soft-edge eigenvalue fluctuations of critical rank-$r$ spiked Wishart and GO/U/SE random matrices. As a corollary of our main result, we prove that if $V$'s growth is at least linear (this includes the multivariate SAO), then $\widehat{H}$'s spectrum is number rigid in the sense of Ghosh and Peres (Duke Math. J., 166(10):1789-1858, 2017). Together with the rigidity of the scalar SAO, this completes the characterization of number rigidity in the soft-edge limits of Gaussian $β$-ensembles and their finite-rank spiked versions.

math.PR↗

arXiv2024

Probabilistic Representations of Ordered Exponentials: Vector-Valued Schrödinger Semigroups and the Combinatorics of Anderson Localization

We provide two applications of an elementary (yet seemingly unknown) probabilistic representation of matrix ordered exponentials, which generalizes the Feynman-Kac formula in finite dimensions and the change of measure formula between two continuous-time Markov processes on a finite state space. Our first and main application consists of a new Feynman-Kac formula for a class of vector-valued Schrödinger operators on the line, which is driven by two sources of randomness: The usual Brownian motion, and a continuous-time Markov process on a finite state space. An important feature of these formulas -- which is at the core of our motivation -- is that they enable the calculation of the joint moments of the semigroup kernels when the matrix potential function contains a continuous Gaussian noise. In particular, our moment formulas shed new light on what the joint moments of the Feynman-Kac kernels of the multivariate stochastic Airy operators of Bloemendal and Virág (Ann. Probab., 44(4):2726--2769, 2016.) should be; we state a precise conjecture to that effect, which we pursue in a forthcoming paper. Our second application consists of Feynman-Kac formulas for the expected square modulus $\mathbf E\big[|Ψ(t,x)|^2\big]$ of the solutions of the Schrödinger equation $\partial_tΨ=-\mathsf i\mathcal H(t)Ψ$ with a time-dependent Hamiltonian $\mathcal H(t)$. Using this, we show that when we take $\mathcal H(t)=-Δ+q(t,x)$ restricted to a finite box within $\mathbb Z^d$, where $q(t,x)$ is a possibly time-dependent Gaussian process, $\mathbf E\big[|Ψ(t,x)|^2\big]$ can be written as a relatively simple expectation that involves self- and mutual-intersections of random walks. In particular, this formula hints at a unified combinatorial mechanism that explains the occurrence of localization for both time-dependent and time-independent noises.

math.PR↗

arXiv2024

Exact Multi-Point Correlations in the Stochastic Heat Equation for Strictly Sublinear Coordinates

We consider the Stochastic Heat Equation (SHE) in $(1+1)$ dimensions with delta Dirac initial data and spacetime white noise. We prove exact large-time asymptotics for multi-point correlations of the SHE for strictly sublinear space coordinates. The sublinear condition is optimal, in the sense that different asymptotics are known to occur when the space coordinates grow linearly [Lin 2023, Theorem 1.1]. Lastly, a notable feature of our result is that it confirms the connection between multi-point correlations in the SHE and the ground state of the Hamiltonian of the delta-Bose gas.

math.PR↗

arXiv2023

Moment Intermittency in the PAM with Asymptotically Singular Noise

Let $ξ$ be a singular Gaussian noise on $\mathbb R^d$ that is either white, fractional, or with the Riesz covariance kernel; in particular, there exists a scaling parameter $ω>0$ such that $c^{ω/2}ξ(c\cdot)$ is equal in distribution to $ξ$ for all $c>0$. Let $(ξ_\varepsilon)_{\varepsilon>0}$ be a sequence of smooth mollifications such that $ξ_\varepsilon\toξ$ as $\varepsilon\to0$. We study the asymptotics of the moments of the parabolic Anderson model (PAM) with noise $ξ_\varepsilon$ as $\varepsilon\to0$, both for large (i.e., $t\to\infty$) and fixed times $t$. This approach makes it possible to study the moments of the PAM with regular and singular noises in a unified fashion, as well as interpolate between the two settings. As corollaries of our main results, we obtain the following: $\textbf{(1)}$ When $ξ$ is subcritical (i.e., $0<ω<2$), our results extend the known large-time moment and tail asymptotics for the Stratonovich PAM with noise $ξ$. Our method of proof clarifies the role of the maximizers of the variational problems (known as Hartree ground states) that appear in these moment asymptotics in describing the geometry of intermittency. We take this opportunity to prove the existence and study the properties of the Hartree ground state with a fractional kernel, which we believe is of independent interest. $\textbf{(2)}$ When $ξ$ is critical or supercritical (i.e., $ω=2$ or $ω>2$), our results provide a new interpretation of the moment blowup phenomenon observed in the Stratonovich PAM with noise $ξ$. That is, we uncover that the latter is related to an intermittency effect that occurs in the PAM with noise $ξ_\varepsilon$ as $\varepsilon\to0$ for $\textit{fixed finite times}$ $t>0$.

math.PR↗

arXiv2023

KPZ equation with a small noise, deep upper tail and limit shape

In this paper, we consider the KPZ equation under the weak noise scaling. That is, we introduce a small parameter $\sqrt{\varepsilon}$ in front of the noise and let $\varepsilon \to 0$. We prove that the one-point large deviation rate function has a $\frac{3}{2}$ power law in the deep upper tail. Furthermore, by forcing the value of the KPZ equation at a point to be very large, we prove a limit shape of the KPZ equation as $\varepsilon \to 0$. This confirms the physics prediction in Kolokolov and Korshunov (2007), Kolokolov and Korshunov (2009), Meerson, Katzav, and Vilenkin (2016), Kamenev, Meerson, and Sasorov (2016), and Le Doussal, Majumdar, Rosso, and Schehr (2016).

math.PR↗

arXiv2023

On Spatial Conditioning of the Spectrum of Discrete Random Schrödinger Operators

Consider a random Schrödinger-type operator of the form $H:=-H_X+V+ξ$ acting on a general graph $\mathscr G=(\mathscr V,\mathscr E)$, where $H_X$ is the generator of a Markov process $X$ on $\mathscr G$, $V$ is a deterministic potential with sufficient growth (so that $H$ has a purely discrete spectrum), and $ξ$ is a random noise with at-most-exponential tails. We prove that $H$'s eigenvalue point process is number rigid in the sense of Ghosh and Peres (Duke Math. J. 166 (2017), no. 10, 1789--1858); that is, the number of eigenvalues in any bounded domain $B\subset\mathbb C$ is determined by the configuration of eigenvalues outside of $B$. Our general setting allows to treat cases where $X$ could be non-symmetric (hence $H$ is non-self-adjoint) and $ξ$ has long-range dependence. Our strategy of proof consists of controlling the variance of the trace of the semigroup $\mathrm e^{-t H}$ using the Feynman-Kac formula.

math-ph↗

arXiv2022

Rigidity of the Stochastic Airy Operator

We prove that the spectrum of the stochastic Airy operator is rigid in the sense of Ghosh and Peres (Duke Math. J., 166(10):1789--1858, 2017) for Dirichlet and Robin boundary conditions. This proves the rigidity of the Airy-$β$ point process and the soft-edge limit of rank-$1$ perturbations of Gaussian $β$-Ensembles for any $β>0$, and solves an open problem mentioned in a previous work of Bufetov, Nikitin, and Qiu (Mosc. Math. J., 19(2):217--274, 2019). Our proof uses a combination of the semigroup theory of the stochastic Airy operator and the techniques for studying insertion and deletion tolerance of point processes developed by Holroyd and Soo (Electron. J. Probab., 18:no. 74, 24, 2013).

math.PR↗

arXiv2022

Asymptotic Freeness of Unitary Matrices in Tensor Product Spaces for Invariant States

In this paper, we pursue our study of asymptotic properties of families of random matrices that have a tensor structure. In previous work, the first- and second-named authors provided conditions under which tensor products of unitary random matrices are asymptotically free with respect to the normalized trace. Here, we extend this result by proving that asymptotic freeness of tensor products of Haar unitary matrices holds with respect to a significantly larger class of states. Our result relies on invariance under the symmetric group, and therefore on traffic probability. As a byproduct, we explore two additional generalisations: (i) we state results of freeness in a context of general sequences of representations of the unitary group -- the fundamental representation being a particular case that corresponds to the classical asymptotic freeness result for Haar unitary matrices, and (ii) we consider actions of the symmetric group and the free group simultaneously and obtain a result of asymptotic freeness in this context as well.

math.PR↗

arXiv2021

Semigroups for One-Dimensional Schrödinger Operators with Multiplicative Gaussian Noise

Let $ H:=-\tfrac12Δ+V$ be a one-dimensional continuum Schrödinger operator. Consider ${\hat H}:= H+ξ$, where $ξ$ is a translation invariant Gaussian noise. Under some assumptions on $ξ$, we prove that if $V$ is locally integrable, bounded below, and grows faster than $\log$ at infinity, then the semigroup $\mathrm e^{-t {\hat H}}$ is trace class and admits a probabilistic representation via a Feynman-Kac formula. Our result applies to operators acting on the whole line $\mathbb R$, the half line $(0,\infty)$, or a bounded interval $(0,b)$, with a variety of boundary conditions. Our method of proof consists of a comprehensive generalization of techniques recently developed in the random matrix theory literature to tackle this problem in the special case where ${\hat H}$ is the stochastic Airy operator.

math.PR↗

arXiv2021

Phase Transitions in Asymptotically Singular Anderson Hamiltonian and Parabolic Model

Let $ξ$ be a Gaussian white noise on $\mathbb R^d$ ($d=1,2,3$). Let $(ξ_\varepsilon)_{\varepsilon>0}$ be continuous Gaussian processes such that $ξ_\varepsilon\toξ$ as $\varepsilon\to0$, defined by convolving $ξ$ against a mollifier. We consider the asymptotics of the parabolic Anderson model (PAM) with noise $ξ_{\varepsilon(t)}$ for large time $t\gg1$, and the Dirichlet eigenvalues of the Anderson Hamiltonian (AH) with potential $ξ_{\varepsilon(t)}$ on large boxes $(-t,t)^d$, where the parameter $\varepsilon(t)$ vanishes as $t\to\infty$. We prove that the asymptotics in question exhibit a phase transition in the rate at which $\varepsilon(t)$ vanishes, which distinguishes between the behavior observed in the AH/PAM with continuous Gaussian noise and white noise. By comparing our main theorems with previous results on the AH/PAM with white noise, our results show that some asymptotics of the latter can be accessed with solely elementary methods, and we obtain quantitative estimates on the difference between the AH/PAM with white noise and its continuous-noise approximations as $t\to\infty$.

math.PR↗

arXiv2020

Spectral rigidity of random Schrödinger operators via Feynman-Kac formulas

We develop a technique for proving number rigidity (in the sense of Ghosh-Peres) of the spectrum of general random Schrödinger operators (RSOs). Our method makes use of Feynman-Kac formulas to estimate the variance of exponential linear statistics of the spectrum in terms of self-intersection local times. Inspired by recent results concerning Feynman-Kac formulas for RSOs with multiplicative white noise by Gorin, Shkolnikov and the first-named author, we use this method to prove number rigidity for a class of one-dimensional continuous RSOs of the form $-\frac12Δ+V+ξ$, where $V$ is a deterministic potential and $ξ$ is a stationary Gaussian noise. Our results require only very mild assumptions on the domain on which the operator is defined, the boundary conditions on that domain, the regularity of the potential $V$, and the singularity of the noise $ξ$.

math-ph↗

arXiv2020

On the Convergence of Random Tridiagonal Matrices to Stochastic Semigroups

We develop an improved version of the stochastic semigroup approach to study the edge of $β$-ensembles pioneered by Gorin and Shkolnikov, and later extended to rank-one additive perturbations by the author and Shkolnikov. Our method is applicable to a significantly more general class of random tridiagonal matrices than that considered in these previous works, including some non-symmetric cases that are not covered by the stochastic operator formalism of Bloemendal, Ramírez, Rider, and Virág. We present two applications of our main results: Firstly, we prove the convergence of $β$-Laguerre-type (i.e., sample covariance) random tridiagonal matrices to the stochastic Airy semigroup and its rank-one spiked version. Secondly, we prove the convergence of the eigenvalues of a certain class of non-symmetric random tridiagonal matrices to the spectrum of a continuum Schrödinger operator with Gaussian white noise potential.

math.PR↗

arXiv2019

Joint and Conditional Local Limit Theorems for Lattice Random Walks and Their Occupation Measures

Let $S_n$ be a lattice random walk with mean zero and finite variance, and let $Λ^a_n$ be its occupation measure at level $a$. In this note, we prove local limit theorems for $\Pr[S_n=x,Λ^a_n=\ell]$ and $\Pr[S_n=x|Λ^a_n=\ell]$ in the cases where $a$, $|x-a|$ and $\ell$ are either zero or at least of order $\sqrt n$. The asymptotic description of these quantities matches the corresponding probabilities for Brownian motion and its local time process. This note can be seen as a generalization of previous results by Kaigh (1975) and Uchiyama (2011). In similar fashion to these results, our method of proof relies on path decompositions that reduce the problem at hand to the study of random walks with independent increments.

math.PR↗