Rooted trees and moments of large sparse random matrices

Oleksiy Khorunzhiy

doi:10.46298/dmtcs.3346

Abstract

In these expository paper we describe the role of the rooted trees as a base for convenient tools in studies ofrandom matrices. Regarding the Wigner ensemble of random matrices, we represent main ingredients ofthis approach. Also werefine our previous result on the limit of the spectral norm of adjacency matrix of large random graphs.

Highlights

∞ 2k is proportional to the number of these walks. Later this description was combined with the graph theory tools to study the spectral norm of large random matrices of this class known as the Wigner ensemble [BY88, FK81]
The use of the graph theory is possible here due to the one-to-one correspondence between the simple half-plane random walks and the set of rooted trees Tk with k edges drawn in the upper half-plane
Another version of the random walks representation is used to prove the universal character of extreme eigenvalue statistics of large random matrices of the Wigner ensemble [Sos99]

Summary

Introduction

The use of the graph theory is possible here due to the one-to-one correspondence between the simple half-plane random walks and the set of rooted trees Tk with k edges drawn in the upper half-plane Another version of the random walks representation is used to prove the universal character of extreme eigenvalue statistics of large random matrices of the Wigner ensemble [Sos99]. In paper [Kho01] it was shown that the trees still represent a simple and convenient description of the corrections to Namely, it was proved that the rooted trees added by the procedure of vertex gluing and shift of cycles describe all terms of 1 ¥ On this way one can separate two different classes of graphs obtained from trees:. We present refinements of results of [Kho01] on this subject

Moments of Wigner random matrices

Moments of sparse random matrices

Discussion