arXiv subjects

Kevin Schnelli

Publications and source records attributed to Kevin Schnelli.

27 records · Page 2Linked to original sources

arXiv2016

Local law of addition of random matrices on optimal scale

The eigenvalue distribution of the sum of two large Hermitian matrices, when one of them is conjugated by a Haar distributed unitary matrix, is asymptotically given by the free convolution of their spectral distributions. We prove that this convergence also holds locally in the bulk of the spectrum, down to the optimal scales larger than the eigenvalue spacing. The corresponding eigenvectors are fully delocalized. Similar results hold for the sum of two real symmetric matrices, when one is conjugated by a Haar orthogonal matrix.

math.PR↗

arXiv2016

Local Stability of the Free Additive Convolution

We prove that the system of subordination equations, defining the free additive convolution of two probability measures, is stable away from the edges of the support and blow-up singularities by showing that the recent smoothness condition of Kargin is always satisfied. As an application, we consider the local spectral statistics of the random matrix ensemble $A+UBU^*$, where $U$ is a Haar distributed random unitary or orthogonal matrix, and $A$ and $B$ are deterministic matrices. In the bulk regime, we prove that the empirical spectral distribution of $A+UBU^*$ concentrates around the free additive convolution of the spectral distributions of $A$ and $B$ on scales down to $N^{-2/3}$.

math.PR↗

arXiv2015

Tracy-Widom Distribution for the Largest Eigenvalue of Real Sample Covariance Matrices with General Population

We consider sample covariance matrices of the form $\mathcal{Q}=(Σ^{1/2}X)(Σ^{1/2} X)^*$, where the sample $X$ is an $M\times N$ random matrix whose entries are real independent random variables with variance $1/N$ and where $Σ$ is an $M\times M$ positive-definite deterministic matrix. We analyze the asymptotic fluctuations of the largest rescaled eigenvalue of $\mathcal{Q}$ when both $M$ and $N$ tend to infinity with $N/M\to d\in(0,\infty)$. For a large class of populations $Σ$ in the sub-critical regime, we show that the distribution of the largest rescaled eigenvalue of $\mathcal{Q}$ is given by the type-1 Tracy-Widom distribution under the additional assumptions that (1) either the entries of $X$ are i.i.d. Gaussians or (2) that $Σ$ is diagonal and that the entries of $X$ have a subexponential decay.

math.PR↗

arXiv2015

Universality for Random Matrix Flows with Time-dependent Density

We show that the Dyson Brownian Motion exhibits local universality after a very short time assuming that local rigidity and level repulsion hold. These conditions are verified, hence bulk spectral universality is proven, for a large class of Wigner-like matrices, including deformed Wigner ensembles and ensembles with non-stochastic variance matrices whose limiting densities differ from the Wigner semicircle law.

math.PR↗

arXiv2014

Edge Universality for Deformed Wigner Matrices

We consider $N\times N$ random matrices of the form $H = W + V$ where $W$ is a real symmetric Wigner matrix and $V$ a random or deterministic, real, diagonal matrix whose entries are independent of $W$. We assume subexponential decay for the matrix entries of $W$ and we choose $V$ so that the eigenvalues of $W$ and $V$ are typically of the same order. For a large class of diagonal matrices $V$ we show that the rescaled distribution of the extremal eigenvalues is given by the Tracy-Widom distribution $F_1$ in the limit of large $N$. Our proofs also apply to the complex Hermitian setting, i.e., when $W$ is a complex Hermitian Wigner matrix.

math.PR↗

arXiv2014

Extremal Eigenvalues and Eigenvectors of Deformed Wigner Matrices

We consider random matrices of the form $H = W + λV$, $λ\in\mathbb{R}^+$, where $W$ is a real symmetric or complex Hermitian Wigner matrix of size $N$ and $V$ is a real bounded diagonal random matrix of size $N$ with i.i.d.\ entries that are independent of $W$. We assume subexponential decay for the matrix entries of $W$ and we choose $λ\sim 1$, so that the eigenvalues of $W$ and $λV$ are typically of the same order. Further, we assume that the density of the entries of $V$ is supported on a single interval and is convex near the edges of its support. In this paper we prove that there is $λ_+\in\mathbb{R}^+$ such that the largest eigenvalues of $H$ are in the limit of large $N$ determined by the order statistics of $V$ for $λ>λ_+$. In particular, the largest eigenvalue of $H$ has a Weibull distribution in the limit $N\to\infty$ if $λ>λ_+$. Moreover, for $N$ sufficiently large, we show that the eigenvectors associated to the largest eigenvalues are partially localized for $λ>λ_+$, while they are completely delocalized for $λ<λ_+$. Similar results hold for the lowest eigenvalues.

math.PR↗

arXiv2013

Local deformed semicircle law and complete delocalization for Wigner matrices with random potential

We consider Hermitian random matrices of the form $H = W + λV$, where $W$ is a Wigner matrix and $V$ a diagonal random matrix independent of $W$. We assume subexponential decay for the matrix entries of $W$ and we choose $λ\sim 1$ so that the eigenvalues of $W$ and $λV$ are of the same order in the bulk of the spectrum. In this paper, we prove for a large class of diagonal matrices $V$ that the local deformed semicircle law holds for $H$, which is an analogous result to the local semicircle law for Wigner matrices. We also prove complete delocalization of eigenvectors and other results about the positions of eigenvalues.

math.PR↗

arXiv2013

Quantum Diffusion with Drift and the Einstein Relation I

We study the dynamics of a quantum particle hopping on a simple cubic lattice and driven by a constant external force. It is coupled to an array of identical, independent thermal reservoirs consisting of free, massless Bose fields, one at each site of the lattice. When the particle visits a site x of the lattice it can emit or absorb field quanta of the reservoir at x. Under the assumption that the coupling between the particle and the reservoirs and the driving force are sufficiently small, we establish the following results: The ergodic average over time of the state of the particle approaches a non-equilibrium steady state (NESS) describing a non-zero mean drift of the particle. Its motion around the mean drift is diffusive, and the diffusion constant and the drift velocity are related to one another by the Einstein relation.

math-ph↗

arXiv2013

Quantum Diffusion with Drift and the Einstein Relation II

This paper is a companion to 'Quantum Diffusion with Drift and the Einstein Relation I' (jointly submitted to arXiv). Its purpose is to describe and prove a certain number of technical results used in 'Quantum Diffusion with Drift and the Einstein Relation I', but not proven there. Both papers concern long-time properties (diffusion, drift) of the motion of a driven quantum particle coupled to an array of thermal reservoirs. The main technical results derived in the present paper are $(1)$ an asymptotic perturbation theory applicable for small driving, and, $(2)$ the construction of time-dependent correlation functions of particle observables.

math-ph↗