Bivariate exponential integrals and edge-bicolored graphs

TL;DR

This paper connects exponential bivariate integrals with generating functions for edge-bicolored graphs, deriving asymptotic formulas and analyzing phase transitions in the Ising model on random graphs.

Contribution

It introduces a novel link between integrals and graph enumeration, providing asymptotic analysis and applications to statistical physics models.

Findings

Asymptotic formulas for regular edge-bicolored graphs

Critical points of polynomials govern asymptotic behavior

Application to phase transitions in the Ising model

Abstract

We show that specific exponential bivariate integrals serve as generating functions of labeled edge-bicolored graphs. Based on this, we prove an asymptotic formula for the number of regular edge-bicolored graphs with arbitrary weights assigned to different vertex structures. The asymptotic behavior is governed by the critical points of a polynomial. As an application, we discuss the Ising model on a random 4-regular graph and show how its phase transitions arise from our formula.