Distribution of Singular Values of Random Band Matrices; Marchenko–Pastur Law and More

Abstract

We consider the limiting spectral distribution of matrices of the form 1 2bn +1 (R + X)(R + X) * , where X is an n × n band matrix of bandwidth bn and R is a non random band matrix of bandwidth bn. We show that the Stieltjes transform of ESD of such matrices converges to the Stieltjes transform of a non-random measure. And the limiting Stieltjes transform satisfies an integral equation. For R = 0, the integral equation yields the Stieltjes transform of the Marchenko-Pastur law.

Circular Law for Random Block Band Matrices with Genuinely Sublinear Bandwidth

“…We follow the proof strategy from [49]. This previous work demonstrated the convergence of the Stieltjes transform for band matrices rather than block band matrices so we necessarily make some adaptations.…”

“…For instance, random band matrices have been studied in the context of nuclear physics, quantum chaos, and systems of interacting particles [23,44,45,76]. Many results have been established for the eigenvalues and eigenvectors of random band matrices, especially Hermitian models; we refer the reader to [4,8,9,11,12,18,21,22,23,31,33,34,35,36,47,48,49,50,51,53,57,58,62,68,69,70,72,79] and references therein.…”

“…The proof proceeds via Girko's Hermitization procedure (see [16]) which is now a standard technique in the study of non-Hermitian random matrices. Following [49], we study the empirical eigenvalue distribution of X z X * z for z ∈ C. In particular, we establish a rate of convergence for the Stieltjes transform X z X * z to the Stieltjes transform of the limiting measure in Section 3. The key technical tool in our proof is a lower bound on the least singular value of X z presented in Section 2 .…”

Modelling equilibration of local many-body quantum systems by random graph ensembles

“…Variants of Wigner band matrices include power-law banded random matrices [39,40] where the standard deviation of matrix elements decreases with the distance from the main diagonal according to a power law, and sparse random band matrices [41], which also account for the possibility of having vanishing elements in the band. However, while the mentioned ensembles of banded random matrices may model certain aspects of physical systems more realistically, they come with the disadvantage that analytic calculations of their spectral properties are far more intricate [42][43][44]. An overview of applications of embedded and banded ensembles is provided in Ref.…”

Modelling equilibration of local many-body quantum systems by random graph ensembles

“…Variants of Wigner band matrices include power-law banded random matrices [39,40] where the standard deviation of matrix elements decreases with the distance from the main diagonal according to a power law, and sparse random band matrices [41], which also account for the possibility of having vanishing elements in the band. However, while the mentioned ensembles of banded random matrices may model certain aspects of physical systems more realistically, they come with the disadvantage that analytic calculations of their spectral properties are far more intricate [42][43][44]. An overview of applications of embedded and banded ensembles is provided in Ref.…”