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Romain Allez

Publications and source records attributed to Romain Allez.

At least 19 recordsLinked to original sources
arXiv2025

The Eigenvector Bead Process

We investigate the overlap matrix between the eigenvectors of a Wigner matrix $H_{N+K}$ of size $(N+K)\times(N+K)$ and those of its principal minor $H_N$ of size $N\times N$, for both the real symmetric ($\beta=1$) and complex Hermitian ($\beta=2$) ensembles, in the regime where $N \to \infty$ while $K$ remains fixed. Our analysis yields two main results. (i) In the \emph{bulk} of the spectrum, an eigenvector of $H_{N+K}$ associated with an eigenvalue at energy level $E$ projects primarily onto eigenvectors of $H_N$ located at the same local spectral level. This phenomenon, which we call \emph{local projection}, highlights a robust stability of the eigenbasis under matrix growth. (ii) At the \emph{spectral edge}, the change of basis between the leading eigenspaces of consecutive minors is asymptotically governed by a random antisymmetric perturbation of order $N^{-1/3}$. In both cases, we provide the asymptotic law of the overlaps expressed in terms of the Airy and Sine kernels. We further extend our analysis to the case of Wishart matrices, that is, sample covariance matrices of the form $W = X^{\!\top} X$, where $X \in \mathbb{R}^{T \times N}$ is a matrix with i.i.d.\ random entries. We establish analogous results for the overlaps between eigenvectors of consecutive minors of $W$, both in the bulk and at the spectral edges (soft and hard). The limiting laws share the same universal structure as in the Wigner case, up to explicit constants depending on the aspect ratio $q = N/T$. This demonstrates the universality of the eigenvector overlap process across distinct random matrix ensembles.

math.PR↗
arXiv2025

Eigenvector Overlaps of Random Covariance Matrices and their Submatrices

We consider the singular vectors of any $m \times n$ submatrix of a rectangular $M \times N$ Gaussian matrix and study their asymptotic overlaps with those of the full matrix, in the macroscopic regime where $N \,/\, M\,$, $m \,/\, M$ as well as $n \,/\, N$ converge to fixed ratios. Our method makes use of the dynamics of the singular vectors and of specific resolvents when the matrix coefficients follow Brownian trajectories. We obtain explicit forms for the limiting rescaled mean squared overlaps for right and left singular vectors in the bulk of both spectra, for any initial matrix $A\,$. When it is null, this corresponds to the Marchenko-Pastur setup for covariance matrices, and our formulas simplify into Cauchy-like functions.

math.PR↗
arXiv2024

Interlacing Eigenvectors of Large Gaussian Matrices

We consider the eigenvectors of the principal minor of dimension $n< N$ of the Dyson Brownian motion in $\mathbb{R}^{N}$ and investigate their asymptotic overlaps with the eigenvectors of the full matrix in the limit of large dimension. We explicitly compute the limiting rescaled mean squared overlaps in the large $n\,, N$ limit with $n\,/\,N$ tending to a fixed ratio $q\,$, for any initial symmetric matrix $A\,$. This is accomplished using a Burgers-type evolution equation for a specific resolvent. In the GOE case, our formula simplifies, and we identify an eigenvector analogue of the well-known interlacing of eigenvalues. We investigate in particular the case where $A$ has isolated eigenvalues. Our method is based on analysing the eigenvector flow under the Dyson Brownian motion.

math.PR↗
arXiv2016

Rotational invariant estimator for general noisy matrices

We investigate the problem of estimating a given real symmetric signal matrix $\textbf{C}$ from a noisy observation matrix $\textbf{M}$ in the limit of large dimension. We consider the case where the noisy measurement $\textbf{M}$ comes either from an arbitrary additive or multiplicative rotational invariant perturbation. We establish, using the Replica method, the asymptotic global law estimate for three general classes of noisy matrices, significantly extending previously obtained results. We give exact results concerning the asymptotic deviations (called overlaps) of the perturbed eigenvectors away from the true ones, and we explain how to use these overlaps to "clean" the noisy eigenvalues of $\textbf{M}$. We provide some numerical checks for the different estimators proposed in this paper and we also make the connection with some well known results of Bayesian statistics.

cond-mat.stat-mech↗
arXiv2015

The continuous Anderson hamiltonian in dimension two

We define the Anderson hamiltonian on the two dimensional torus $\mathbb R^2/\mathbb Z^2$. This operator is formally defined as $\mathscr H:= -Δ+ ξ$ where $Δ$ is the Laplacian operator and where $ξ$ belongs to a general class of singular potential which includes the Gaussian white noise distribution. We use the notion of paracontrolled distribution as introduced by Gubinelli, Imkeller and Perkowski in [14]. We are able to define the Schrödinger operator $\mathscr H$ as an unbounded self-adjoint operator on $L^2(\mathbb T^2)$ and we prove that its real spectrum is discrete with no accumulation points for a general class of singular potential $ξ$. We also establish that the spectrum is a continuous function of a sort of enhancement $Ξ(ξ)$ of the potential $ξ$. As an application, we prove that a correctly renormalized smooth approximations $\mathscr H_\varepsilon:= -Δ+ ξ_\varepsilon+c_\varepsilon$ (where $ξ_\varepsilon$ is a smooth mollification of the Gaussian white noise $ξ$ and $c_\varepsilon$ an explicit diverging renormalization constant) converge in the sense of the resolvent towards the singular operator $\mathscr H$. In the case of a Gaussian white noise $ξ$, we obtain exponential tail bounds for the minimal eigenvalue (sometimes called ground state) of the operator $\mathscr H$ as well as its order of magnitude $\log L$ when the operator is considered on a large box $\mathbb T_L:= \mathbb R^2/(L\mathbb Z)^2$ with $L\to \infty$.

math.PR↗
arXiv2015

The eigenvectors of Gaussian matrices with an external source

We consider a diffusive matrix process $(X_t)_{t\ge 0}$ defined as $X_t:=A+H_t$ where $A$ is a given deterministic Hermitian matrix and $(H_t)_{t\ge 0}$ is a Hermitian Brownian motion. The matrix $A$ is the "external source" that one would like to estimate from the noisy observation $X_t$ at some time $t>0$. We investigate the relationship between the non-perturbed eigenvectors of the matrix $A$ and the perturbed eigenstates at some time $t$ for the three relevant scaling relations between the time $t$ and the dimension $N$ of the matrix $X_t$. We determine the asymptotic (mean-squared) projections of any given non-perturbed eigenvector $|ψ_j^0\rangle$, associated to an eigenvalue $a_j$ of $A$ which may lie inside the bulk of the spectrum or be isolated (spike) from the other eigenvalues, on the orthonormal basis of the perturbed eigenvectors $|ψ_i^t\rangle,i\neq j$. We derive a Burgers type evolution equation for the local resolvent $(z-X_t)_{ii}^{-1}$, describing the evolution of the local density of a given initial state $|ψ_j ^0\rangle$. We are able to solve this equation explicitly in the large $N$ limit, for any initial matrix $A$. In the case of one isolated eigenvector $|ψ_j^0\rangle$, we prove a central limit Theorem for the overlap $\langle ψ_j^0|ψ_j^t\rangle$. When properly centered and rescaled by a factor $\sqrt{N}$, this overlap converges in law towards a centered Gaussian distribution with an explicit variance depending on $t$. Our method is based on analyzing the eigenvector flow under the Dyson Brownian motion.

math.PR↗
arXiv2015

Eigenvector dynamics under free addition

We investigate the evolution of a given eigenvector of a symmetric (deterministic or random) matrix under the addition of a matrix in the Gaussian orthogonal ensemble. We quantify the overlap between this single vector with the eigenvectors of the initial matrix and identify precisely a "Cauchy-flight" regime. In particular, we compute the local density of this vector in the eigenvalues space of the initial matrix. Our results are obtained in a non perturbative setting and are derived using the ideas of [O. Ledoit and S. Péché, Prob. Th. Rel. Fields, {\bf 151} 233 (2011)]. Finally, we give a robust derivation of a result obtained in [R. Allez and J.-P. Bouchaud, Phys. Rev. E {\bf 86}, 046202 (2012)] to study eigenspace dynamics in a semi-perturbative regime.

math.PR↗
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↗
arXiv2014

Index Distribution of the Ginibre Ensemble

Complex systems, and in particular random neural networks, are often described by randomly interacting dynamical systems with no specific symmetry. In that context, characterizing the number of relevant directions necessitates fine estimates on the Ginibre ensemble. In this Letter, we compute analytically the probability distribution of the number of eigenvalues $N_R$ with modulus greater than $R$ (the index) of a large $N\times N$ random matrix in the real or complex Ginibre ensemble. We show that the fraction $N_R/N=p$ has a distribution scaling as $\exp(-βN^2 ψ_R(p))$ with $β=1$ (respectively $β=1/2$) for the complex (resp. real) Ginibre ensemble. For any $p\in[0,1]$, the equilibrium spectral densities as well as the rate function $ψ_R(p)$ are explicitly derived. This function displays a third order phase transition at the critical (minimum) value $p^*_R=1-R^2$, associated to a phase transition of the Coulomb gas. We deduce that, in the central regime, the fluctuations of the index $N_R$ around its typical value $p^*_R N$ scale as $N^{1/3}$.

math.PR↗
arXiv2012

Invariant $β$-Wishart ensembles, crossover densities and asymptotic corrections to the Marchenko-Pastur law

We construct a diffusive matrix model for the $β$-Wishart (or Laguerre) ensemble for general continuous $β\in [0,2]$, which preserves invariance under the orthogonal/unitary group transformation. Scaling the Dyson index $β$ with the largest size $M$ of the data matrix as $β=2c/M$ (with $c$ a fixed positive constant), we obtain a family of spectral densities parametrized by $c$. As $c$ is varied, this density interpolates continuously between the Mar\vcenko-Pastur ($c\to \infty$ limit) and the Gamma law ($c\to 0$ limit). Analyzing the full Stieltjes transform (resolvent) equation, we obtain as a byproduct the correction to the Mar\vcenko-Pastur density in the bulk up to order 1/M for all $β$ and up to order $1/M^2$ for the particular cases $β=1,2$.

cond-mat.stat-mech↗
arXiv2012

Lognormal scale invariant random measures

In this article, we consider the continuous analog of the celebrated Mandelbrot star equation with lognormal weights. Mandelbrot introduced this equation to characterize the law of multiplicative cascades. We show existence and uniqueness of measures satisfying the aforementioned continuous equation; these measures fall under the scope of the Gaussian multiplicative chaos theory developed by J.P. Kahane in 1985 (or possibly extensions of this theory). As a by product, we also obtain an explicit characterization of the covariance structure of these measures. We also prove that qualitative properties such as long-range independence or isotropy can be read off the equation.

math.PR↗
arXiv2012

Eigenvector dynamics: general theory and some applications

We propose a general framework to study the stability of the subspace spanned by $P$ consecutive eigenvectors of a generic symmetric matrix ${\bf H}_0$, when a small perturbation is added. This problem is relevant in various contexts, including quantum dissipation (${\bf H}_0$ is then the Hamiltonian) and financial risk control (in which case ${\bf H}_0$ is the assets return covariance matrix). We argue that the problem can be formulated in terms of the singular values of an overlap matrix, that allows one to define a "fidelity" distance. We specialize our results for the case of a Gaussian Orthogonal ${\bf H}_0$, for which the full spectrum of singular values can be explicitly computed. We also consider the case when ${\bf H}_0$ is a covariance matrix and illustrate the usefulness of our results using financial data. The special case where the top eigenvalue is much larger than all the other ones can be investigated in full detail. In particular, the dynamics of the angle made by the top eigenvector and its true direction defines an interesting new class of random processes.

cond-mat.stat-mech↗
arXiv2012

Invariant $β$-ensembles and the Gauss-Wigner crossover

We define a new diffusive matrix model converging towards the $β$ -Dyson Brownian motion for all $β\in [0,2]$ that provides an explicit construction of $β$-ensembles of random matrices that is invariant under the orthogonal/unitary group. For small values of $β$, our process allows one to interpolate smoothly between the Gaussian distribution and the Wigner semi-circle. The interpolating limit distributions form a one parameter family that can be explicitly computed. This also allows us to compute the finite-size corrections to the semi-circle.

math.PR↗
arXiv2012

Marchenko Pastur type theorem for independent MRW processes: convergence of the empirical spectral measure

We study the asymptotic of the spectral distribution for large empirical covariance matrices composed of independent Multifractal Random Walk processes. The asymptotic is taken as the observation lag shrinks to 0. In this setting, we show that there exists a limiting spectral distribution whose Stieltjes transform is uniquely characterized by equations which we specify. We also illustrate our results by numerical simulations.

math.PR↗
arXiv2012

A diffusive matrix model for invariant $β$-ensembles

We define a new diffusive matrix model converging towards the $β$-Dyson Brownian motion for all $β\in [0,2]$ that provides an explicit construction of $β$-ensembles of random matrices that is invariant under the orthogonal/unitary group. We also describe the eigenvector dynamics of the limiting matrix process; we show that when $β< 1$ and that two eigenvalues collide, the eigenvectors of these two colliding eigenvalues fluctuate very fast and take the uniform measure on the orthocomplement of the eigenvectors of the remaining eigenvalues.

math.PR↗
arXiv2011

Eigenvector dynamics: theory and some applications

We propose a general framework to study the stability of the subspace spanned by $P$ consecutive eigenvectors of a generic symmetric matrix ${\bf H}_0$, when a small perturbation is added. This problem is relevant in various contexts, including quantum dissipation (${\bf H}_0$ is then the Hamiltonian) and risk control (in which case ${\bf H}_0$ is the assets return correlation matrix). We specialize our results for the case of a Gaussian Orthogonal ${\bf H}_0$, or when ${\bf H}_0$ is a correlation matrix. We illustrate the usefulness of our framework using financial data.

cond-mat.stat-mech↗