Asymptotic Behavior of Partition Functions with Graph Laplacian

TL;DR

This paper investigates the asymptotic properties of partition functions associated with graph Laplacians, revealing that the large-scale behavior is governed by the enumeration of specific acyclic diagrams in graph models.

Contribution

It introduces a new discrete matrix model with a quartic potential based on graph Laplacians and analyzes its large-n asymptotics.

Findings

Large-n free energy is determined by connected acyclic diagrams.

The model provides a discrete analog of matrix models with quartic potential.

The approach links graph enumeration with statistical mechanics.

Abstract

We introduce the matrix sums that represent a discrete analog of the matrix models with quartic potential. The probability space is given by the set of all simple n-vertex graphs with the Gibbs weight determined by the graph Laplacian. We study the large-n limit of the free energy per site and show that it is determined by the number of connected acyclic diagrams on the set of two-valent vertices.