Spectral properties of Levy Rosenzweig-Porter model via supersymmetric approach

TL;DR

This paper analytically computes the spectral density of Levy and Levy-Rosenzweig-Porter matrices with non-Gaussian heavy-tailed entries using supersymmetric methods, revealing phase transition behavior.

Contribution

It introduces a functional Hubbard-Stratonovich transformation to analytically derive spectral densities for non-Gaussian random matrices, advancing understanding of their phase transitions.

Findings

Spectral density depends on a control parameter indicating phase transition.

The method handles non-Gaussian heavy-tailed matrix entries.

Spectral density serves as an order parameter for ergodic to fractal transition.

Abstract

By using the Efetov's super-symmetric formalism we computed analytically the mean spectral density $\rho(E)$ for the L\'evy and the L\'evy -Rosenzweig-Porter random matrices which off-diagonal elements are strongly non-Gaussian with power-law tails. This makes the standard Hubbard-Stratonovich transformation inapplicable to such problems. We used, instead, the functional Hubbard-Stratonovich transformation which allowed to solve the problem analytically for large sizes of matrices. We show that $\rho(E)$ depends crucially on the control parameter that drives the system through the transition between the ergodic and the fractal phases and it can be used as an order parameter.