On Connected Diagrams and Cumulants of Erdos-Renyi Matrix Models

TL;DR

This paper introduces two new discrete matrix models based on Erdős-Rényi graphs, analyzing their cumulant expansions and limiting behaviors to connect graph walk statistics with random matrix theory.

Contribution

It presents novel discrete matrix models for graph walks and studies their cumulant structures and asymptotic limits, bridging graph theory and random matrix theory.

Findings

Derived characteristic functions for walk counts on random graphs.

Analyzed cumulant expansions and their diagram structures.

Explored limits for different edge probability regimes.

Abstract

Regarding the adjacency matrices of n-vertex graphs and related graph Laplacian, we introduce two families of discrete matrix models constructed both with the help of the Erdos-Renyi ensemble of random graphs. Corresponding matrix sums represent the characteristic functions of the average number of walks and closed walks over the random graph. These sums can be considered as discrete analogs of the matrix integrals of random matrix theory. We study the diagram structure of the cumulant expansions of logarithms of these matrix sums and analyze the limiting expressions in the cases of constant and vanishing edge probabilities as n tends to infinity.