On the localization transition in symmetric random matrices

TL;DR

This paper investigates the localization transition in large symmetric random matrices using the cavity method, focusing on Laplacian and Lévy matrix ensembles, and derives a critical line for Lévy matrices.

Contribution

It introduces a cavity method analysis of localization in large symmetric matrices and derives a critical line for Lévy matrices, advancing understanding of localization phenomena.

Findings

Critical line for localization in Lévy matrices derived

Theoretical results match finite matrix diagonalization

Localization transition characterized for specific ensembles

Abstract

We study the behaviour of the inverse participation ratio and the localization transition in infinitely large random matrices through the cavity method. Results are shown for two ensembles of random matrices: Laplacian matrices on sparse random graphs and fully-connected L\'evy matrices. We derive a critical line separating localized from extended states in the case of L\'evy matrices. Comparison between theoretical results and diagonalization of finite random matrices is shown.