Virial expansion for almost diagonal random matrices

TL;DR

This paper develops a diagrammatic expansion method to analyze energy level statistics of almost diagonal random matrices, providing explicit formulas for spectral form-factors in various ensembles and their crossovers.

Contribution

It introduces a novel diagrammatic technique based on the Trotter formula for calculating spectral form-factors in almost diagonal random matrices, extending analysis to GOE, GUE, and crossover ensembles.

Findings

Derived expressions for K_1(τ) and K_2(τ) in various ensembles.

Provided series expansions for spectral form-factors in almost diagonal matrices.

Analyzed specific ensembles like Rosenzweig-Porter and power-law banded matrices.

Abstract

Energy level statistics of Hermitian random matrices $\hat H$ with Gaussian independent random entries $H_{i\geq j}$ is studied for a generic ensemble of almost diagonal random matrices with $ <|H_{ii}|^{2} > \sim 1$ and $<|H_{i\neq j}|^{2} >= b {\cal F}(|i-j|) \ll 1$. We perform a regular expansion of the spectral form-factor $K(\tau) = 1 + b K_{1}(\tau) + b^{2} K_{2}(\tau) + ... $ in powers of $b \ll 1$ with the coefficients $K_{m}(\tau)$ that take into account interaction of (m+1) energy levels. To calculate $K_{m}(\tau)$, we develop a diagrammatic technique which is based on the Trotter formula and on the combinatorial problem of graph edges coloring with (m+1) colors. Expressions for $K_{1}(\tau)$ and $K_{2}(\tau)$ in terms of infinite series are found for a generic function ${\cal F}(|i-j|)$ in the Gaussian Orthogonal Ensemble (GOE), the Gaussian Unitary Ensemble (GUE) and in the crossover between them (the almost unitary Gaussian ensemble). The Rosenzweig-Porter and power-law banded matrix ensembles are considered as examples.