Asymptotic properties of resolvents of large dilute Wigner matrices

TL;DR

This paper investigates the spectral properties of large dilute Wigner matrices, demonstrating that under certain conditions, their spectral distribution follows the semicircle law and their local spectral statistics match those of classical Wigner matrices, indicating preserved universality.

Contribution

It establishes the validity of the semicircle law and universality of local spectral statistics for dilute Wigner matrices under broad conditions.

Findings

Semicircle law holds for dilute Wigner matrices as n→∞ and p→∞.

Local spectral statistics converge to those of classical Wigner matrices.

Moderate dilution does not change the universality class of the spectral distribution.

Abstract

We study the spectral properties of the dilute Wigner random real symmetric n-dimensional matrices H such that the entries H(i,j) take zero value with probability 1-p/n. We prove that under rather general conditions on the probability distribution of H(i,j) the semicircle law is valid for the dilute Wigner ensemble in the limit of infinite n and p. In the second part of the paper we study the leading term of the correlation function of the resolvent G(z) of H with large enough Im z in the limit of infinite n and p such that 3/5 log n <log p < log n. We show that this leading term, when considered on the local spectral scale, converges to the same limit as that of the resolvent correlation function of the Wigner ensemble of random matrices. This shows that the moderate dilution of the Wigner ensemble does not alter its universality class.