arXiv subjects

Laure Dumaz

Publications and source records attributed to Laure Dumaz.

17 recordsLinked to original sources

arXiv2026

The spectrum of the stochastic Bessel operator at high temperature

Ramírez and Rider (2009) established that the hard edge of the spectrum of the $β$-Laguerre ensemble converges, in the high-dimensional limit, to the bottom of the spectrum of the stochastic Bessel operator. Using stochastic analysis tools, we prove that, in the high-temperature limit ($β\to 0$), the rescaled eigenvalue point process of this operator converges to a non-trivial limiting point process. This limit is characterized by a family of coupled diffusions and differs from a Poisson point process due to its interaction with the hard edge. Exploiting this diffusion characterization, we establish exact large deviation asymptotics for the largest eigenvalues. Furthermore, for an explicit range of the parameters, we relate this limiting process to the finite-$n$ $β$-Laguerre ensemble, conjecturing an exact distributional match with the infinite sum of its independent exponential gaps. As a byproduct of our analysis, we also formulate a conjecture regarding an explicit integral formula for the probability that a reflected Brownian motion with a constant drift hits an affine line, generalizing a formula of Salminen and Yor (2011).

math.PR↗

arXiv2026

Long-Range Correlation of the Sine$_β$ point Process

We study the correlations of the celebrated Sine$_β$ point process. This point process arises as the bulk scaling limit of $β$-ensembles and has a geometric description through the Brownian carousel, as shown by Valkó and Virág (2009). We establish that the averaged $k$-point truncated correlation functions decay polynomially in the limit of large separation. We show that the decay exponent is of order $1/β$ for large $β$. This is a step towards a conjecture by Forrester and Haldane regarding the exact asymptotics of the two-point correlation function, a problem recently addressed by Qu and Valkó (2025). Our proofs, which rely on a careful analysis of the coupling of diffusions associated with the Brownian carousel, hold for all $β>0$ and $k \geq 1$, significantly extending previous results limited to specific values of $β$ or $k$.

math.PR↗

arXiv2022

Anderson localization for the $1$-d Schrödinger operator with white noise potential

We consider the random Schrödinger operator on $\mathbb{R}$ obtained by perturbing the Laplacian with a white noise. We prove that Anderson localization holds for this operator: almost surely the spectral measure is pure point and the eigenfunctions are exponentially localized. We give two separate proofs of this result. We also present a detailed construction of the operator and relate it to the parabolic Anderson model. Finally, we discuss the case where the noise is smoothed out.

math.PR↗

arXiv2021

The delocalized phase of the Anderson Hamiltonian in $1$-d

We introduce a random differential operator, that we call the $\mathtt{CS}_τ$ operator, whose spectrum is given by the $\mbox{Sch}_τ$ point process introduced by Kritchevski, Valkó and Virág (2012) and whose eigenvectors match with the description provided by Rifkind and Virág (2018). This operator acts on $\mathbf{R}^2$-valued functions from the interval $[0,1]$ and takes the form: $$ 2 \begin{pmatrix} 0 & -\partial_t \\ \partial_t & 0 \end{pmatrix} + \sqrtτ \begin{pmatrix} d\mathcal{B} + \frac1{\sqrt 2} d\mathcal{W}_1 & \frac1{\sqrt 2} d\mathcal{W}_2\\ \frac1{\sqrt 2} d\mathcal{W}_2 & d\mathcal{B} - \frac1{\sqrt 2} d\mathcal{W}_1\end{pmatrix}\,, $$ where $d\mathcal{B}$, $d\mathcal{W}_1$ and $d\mathcal{W}_2$ are independent white noises. Then, we investigate the high part of the spectrum of the Anderson Hamiltonian $\mathcal{H}_L := -\partial_t^2 + dB$ on the segment $[0,L]$ with white noise potential $dB$, when $L\to\infty$. We show that the operator $\mathcal{H}_L$, recentred around energy levels $E \sim L/τ$ and unitarily transformed, converges in law as $L\to\infty$ to $\mathtt{CS}_τ$ in an appropriate sense. This allows to answer a conjecture of Rifkind and Virág (2018) on the behavior of the eigenvectors of $\mathcal{H}_L$. Our approach also explains how such an operator arises in the limit of $\mathcal{H}_L$. Finally we show that at higher energy levels, the Anderson Hamiltonian matches (asymptotically in $L$) with the unperturbed Laplacian $-\partial_t^2$. In a companion paper, it is shown that at energy levels much smaller than $L$, the spectrum is localized with Poisson statistics: the present paper therefore identifies the delocalized phase of the Anderson Hamiltonian.

math.PR↗

arXiv2021

Localization crossover for the continuous Anderson Hamiltonian in $1$-d

We investigate the behavior of the spectrum of the continuous Anderson Hamiltonian $\mathcal{H}_L$, with white noise potential, on a segment whose size $L$ is sent to infinity. We zoom around energy levels $E$ either of order $1$ (Bulk regime) or of order $1\ll E \ll L$ (Crossover regime). We show that the point process of (appropriately rescaled) eigenvalues and centers of mass converge to a Poisson point process. We also prove exponential localization of the eigenfunctions at an explicit rate. In addition, we show that the eigenfunctions converge to well-identified limits: in the Crossover regime, these limits are universal. Combined with the results of our companion paper arXiv:2102.05393, this identifies completely the transition between the localized and delocalized phases of the spectrum of $\mathcal{H}_L$. The two main technical challenges are the proof of a two-points or Minami estimate, as well as an estimate on the convergence to equilibrium of a hypoelliptic diffusion, the proof of which relies on Malliavin calculus and the theory of hypocoercivity.

math.PR↗

arXiv2020

The stochastic Airy operator at large temperature

It was shown in [J. A. Ramírez, B. Rider and B. Virág. J. Amer. Math. Soc. 24, 919-944 (2011)] that the edge of the spectrum of $β$ ensembles converges in the large $N$ limit to the bottom of the spectrum of the stochastic Airy operator. In the present paper, we obtain a complete description of the bottom of this spectrum when the temperature $1/β$ goes to $\infty$: we show that the point process of appropriately rescaled eigenvalues converges to a Poisson point process on $\mathbb{R}$ of intensity $e^x dx$ and that the eigenfunctions converge to Dirac masses centered at IID points with exponential laws. Furthermore, we obtain a precise description of the microscopic behavior of the eigenfunctions near their localization centers.

math.PR↗

arXiv2020

Operator level hard-to-soft transition for $β$-ensembles

The soft and hard edge scaling limits of $β$-ensembles can be characterized as the spectra of certain random Sturm-Liouville operators. It has been shown that by tuning the parameter of the hard edge process one can obtain the soft edge process as a scaling limit. We prove that this limit can be realized on the level of the corresponding random operators. More precisely, the random operators can be coupled in a way so that the scaled versions of the hard edge operators converge to the soft edge operator a.s. in the norm resolvent sense.

math.PR↗

arXiv2019

Near-critical spanning forests and renormalization

We study random two-dimensional spanning forests in the plane that can be viewed both in the discrete case and in their appropriately taken scaling limits as a uniformly chosen spanning tree with some Poissonian deletion of edges or points. We show how to relate these scaling limits to a stationary distribution of a natural coalescent-type Markov process on a state-space of abstract graphs with real-valued edge-weights. This Markov process can be interpreted as a renormalization flow. This provides a model for which one can rigorously implement the formalism proposed by the third author in order to relate the law of the scaling limit of a critical model to a stationary distribution of such a renormalization/Markov process: When starting from any two-dimensional lattice with constant edge-weights, the Markov process does indeed converge in law to this stationary distribution that corresponds to a scaling limit of UST with Poissonian deletions. The results of this paper heavily build on the convergence in distribution of branches of the UST to SLE$_2$ (a result by Lawler, Schramm and Werner) as well as on the convergence of the suitably renormalized length of the loop-erased random walk to the "natural parametrization" of the SLE$_2$ (a recent result by Lawler and Viklund).

math.PR↗

arXiv2019

Localization of the continuous Anderson Hamiltonian in $1$-d

We study the bottom of the spectrum of the Anderson Hamiltonian $\mathcal{H}_L := -\partial_x^2 + ξ$ on $[0,L]$ driven by a white noise $ξ$ and endowed with either Dirichlet or Neumann boundary conditions. We show that, as $L\rightarrow\infty$, the point process of the (appropriately shifted and rescaled) eigenvalues converges to a Poisson point process on $\mathbb{R}$ with intensity $e^x dx$, and that the (appropriately rescaled) eigenfunctions converge to Dirac masses located at independent and uniformly distributed points. Furthermore, we show that the shape of each eigenfunction, recentered around its maximum and properly rescaled, is given by the inverse of a hyperbolic cosine. We also show that the eigenfunctions decay exponentially from their localization centers at an explicit rate, and we obtain very precise information on the zeros and local maxima of these eigenfunctions. Finally, we show that the eigenvalues/eigenfunctions in the Dirichlet and Neumann cases are very close to each other and converge to the same limits.

math.PR↗

arXiv2018

Acoustic and geoacoustic inverse problems in randomly perturbed shallow-water environments

The main goal of this paper is to estimate the regional acoustic and geoacoustic shallow-water environment from data collected by a vertical hydrophone array and transmitted by distant time-harmonic point sources. We aim at estimating the statistical properties of the random fluctuations of the index of refraction in the water column and the characteristics of the sea bottom. We first explain from first principles how acoustic wave propagation can be expressed as Markovian dynamics for the complex mode amplitudes of the sound pressure, which makes it possible to express the cross moments of the sound pressure in terms of the parameters to be estimated. We then show how the estimation problem can be formulated as a nonlinear inverse problem using this formulation, that can be solved by minimization of a misfit function. We apply this method to experimental data collected by the ALMA system (Acoustic Laboratory for Marine Applications).

math.AP↗

arXiv2014

From Sine kernel to Poisson statistics

We study the Sine$_β$ process introduced in [B. Valkó and B. Virág. Invent. math. (2009)] when the inverse temperature $β$ tends to 0. This point process has been shown to be the scaling limit of the eigenvalues point process in the bulk of $β$-ensembles and its law is characterized in terms of the winding numbers of the Brownian carrousel at different angular speeds. After a careful analysis of this family of coupled diffusion processes, we prove that the Sine$_β$ point process converges weakly to a Poisson point process on $\mathbb{R}$. Thus, the Sine$_β$ point processes establish a smooth crossover between the rigid clock (or picket fence) process (corresponding to $β=\infty$) and the Poisson process.

math.PR↗

arXiv2014

Tracy-Widom at high temperature

We investigate the marginal distribution of the bottom eigenvalues of the stochastic Airy operator when the inverse temperature $β$ tends to $0$. We prove that the minimal eigenvalue, whose fluctuations are governed by the Tracy-Widom $β$ law, converges weakly, when properly centered and scaled, to the Gumbel distribution. More generally we obtain the convergence in law of the marginal distribution of any eigenvalue with given index $k$. Those convergences are obtained after a careful analysis of the explosion times process of the Riccati diffusion associated to the stochastic Airy operator. We show that the empirical measure of the explosion times converges weakly to a Poisson point process using estimates proved in [L. Dumaz and B. Virág. Ann. Inst. H. Poincaré Probab. Statist. 49, 4, 915-933, (2013)]. We further compute the empirical eigenvalue density of the stochastic Airy ensemble on the macroscopic scale when $β\to 0$. As an application, we investigate the maximal eigenvalues statistics of $β_N$-ensembles when the repulsion parameter $β_N\to 0$ when $N\to +\infty$. We study the double scaling limit $N\to +\infty, β_N \to 0$ and argue with heuristic and numerical arguments that the statistics of the marginal distributions can be deduced following the ideas of [A. Edelman and B. D. Sutton. J. Stat. Phys. 127, 6, 1121-1165 (2007)] and [J. A. Ramírez, B. Rider and B. Virág. J. Amer. Math. Soc. 24, 919-944 (2011)] from our later study of the stochastic Airy operator.

math.PR↗

arXiv2014

Random matrices in non-confining potentials

We consider invariant matrix processes diffusing in non-confining cubic potentials of the form $V_a(x)= x^3/3 - a x, a\in \mathbb{R}$. We construct the trajectories of such processes for all time by restarting them whenever an explosion occurs, from a new (well chosen) initial condition, insuring continuity of the eigenvectors and of the non exploding eigenvalues. We characterize the dynamics of the spectrum in the limit of large dimension and analyze the stationary state of this evolution explicitly. We exhibit a sharp phase transition for the limiting spectral density $ρ_a$ at a critical value $a=a^*$. If $a\geq a^*$, then the potential $V_a$ presents a well near $x=\sqrt{a}$ deep enough to confine all the particles inside, and the spectral density $ρ_a$ is supported on a compact interval. If $a<a^*$ however, the steady state is in fact dynamical with a macroscopic stationary flux of particles flowing across the system. In this regime, the eigenvalues allocate according to a stationary density profile $ρ_{a}$ with full support in $\mathbb{R}$, flanked with heavy tails such that $ρ_{a}(x)\sim C_a /x^2$ as $x\to \pm \infty$. Our method applies to other non-confining potentials and we further investigate a family of quartic potentials, which were already studied in Brézin et al. to count planar diagrams.

math.PR↗

arXiv2012

Marginal densities of the "true" self-repelling motion

Let X(t) be the true self-repelling motion (TSRM) constructed by B.T. and Wendelin Werner in 1998, L(t,x) its occupation time density (local time) and H(t):=L(t,X(t)) the height of the local time profile at the actual position of the motion. The joint distribution of (X(t),H(t)) was identified by B.T. in 1995 in somewhat implicit terms. Now we give explicit formulas for the densities of the marginal distributions of X(t) and H(t). The distribution of X(t) has a particularly surprising shape: It has a sharp local minimum with discontinuous derivative at 0. As a consequence we also obtain a precise version of the large deviation estimate of arXiv:1105.2948v3.

math.PR↗

arXiv2012

Large deviations and path properties of the true self-repelling motion

We derive some large deviation bounds for events related to the "true self-repelling motion", a one-dimensional self-interacting process introduced by Toth and Werner, that has very different path properties than usual diffusion processes. We then use these estimates to study certain of these path properties such as its law of iterated logarithms for both small and large times.

math.PR↗

arXiv2012

A clever (self-repelling) burglar

We derive the following property of the "true self-repelling motion", a continuous real-valued self-interacting process (X_t, t \ge 0) introduced by Balint Toth and Wendelin Werner. Conditionally on its occupation time measure at time one (which is the information about how much time it has spent where before time one), the law of X_1 is uniform in a certain admissible interval. This contrasts with the corresponding conditional distribution for Brownian motion that had been studied by Warren and Yor.

math.PR↗

arXiv2011

The right tail exponent of the Tracy-Widom-beta distribution

The Tracy-Widom beta distribution is the large dimensional limit of the top eigenvalue of beta random matrix ensembles. We use the stochastic Airy operator representation to show that as a tends to infinity the tail of the Tracy Widom distribution satisfies P(TW_beta > a) = a^(-3/4 beta+o(1)) exp(-2/3 beta a^(3/2)).

math.PR↗