Empirical Spectral Distribution of a Matrix Under Perturbation

Abstract

Abstract. We provide a perturbative expansion for the empirical spectral distribution of a Hermitian matrix with large size perturbed by a random matrix with small operator norm whose entries in the eigenvector basis of the first one are independent with a variance profile. We prove that, depending on the order of magnitude of the perturbation, several regimes can appear, called perturbative and semi-perturbative regimes. Depending on the regime, the leading terms of the expansion are either related to the one-dimensional Gaussian free field or to free probability theory.

Eigenstate entanglement between quantum chaotic subsystems: Universal transitions and power laws in the entanglement spectrum

“…Its increase is governed by the limit of C(α)/(α − 1) as α → 1 which is equal to π 3/2 , consistent with Eq. (78). Explicitly the von Neumann entropy is is surprisingly good with Eqs.…”

“…Perturbation theory for random matrix ensembles has previously been applied to describe symmetry breaking [26,38,68,75], and continues to be of interest due to various applications ranging from quantum mechanics to quantitative finance [76][77][78]. Mostly this has been done in a Hamiltonian framework whereas the ensemble of Eq.…”

Eigenvectors of a matrix under random perturbation

“…In contrast with [4], where we studied the empirical spectral measure µ ε n of the matrix D ε n , we consider here the spectral measure µ ε n,e i of D ε n over a vector e i of the canonical basis, defined 2 FLORENT BENAYCH-GEORGES, NATHANAËL ENRIQUEZ, AND ALKÉOS MICHAÏL through an eigenvector basis (u ε j ) j∈{1,...,n} of D ε n and the related eigenvalues (λ ε j ) j∈{1,...,n} by µ ε n,e i := n j=1 | u ε j , e i | 2 δ λ ε j .…”

“…This paper is devoted to the study of the sensitivity of the eigenvectors of a given operator under small perturbations. In the previous paper [4] we studied the effect of a perturbation on the spectrum of a diagonal matrix by a random matrix with small operator norm and whose entries in the eigenvector basis of the first one were independent, centered, with a variance profile. We provided a perturbative expansion of the empirical spectral distribution, but did not consider the deformation of the eigenvectors basis with respect to the canonical basis.…”

Spectral deconvolution of unitarily invariant matrix models

“…Both hypotheses yield the same results but require different tools. In this paper, we focus on the second hypothesis of a unitarily invariant noise, which has already been studied in [BABP16,BGEM19,LP11]. Note that in the case of Gaussian matrices with independent entries, the hypothesis of unitarily invariance of the distribution is also satisfied.…”