Cumulant expansion for counting Eulerian orientations

TL;DR

This paper develops an asymptotic expansion to count Eulerian orientations in certain graphs, using a new tail bound for cumulant expansion, advancing understanding in combinatorics and statistical physics models.

Contribution

It introduces a novel tail bound for cumulant expansion of the Laplace transform and applies it to derive precise asymptotics for counting Eulerian orientations in graphs with good expansion.

Findings

Asymptotic expansion for Eulerian orientations count

New tail bound for cumulant expansion of Laplace transform

Applicable to graphs with degrees larger than log^8 n

Abstract

An Eulerian orientation is an orientation of the edges of a graph such that every vertex is balanced: its in-degree equals its out-degree. Counting Eulerian orientations corresponds to the crucial partition function in so-called ``ice-type models'' in statistical physics and is known to be hard for general graphs. For all graphs with good expansion properties and degrees larger than $\log^{8} n$, we derive an asymptotic expansion for this count that approximates it to precision $O(n^{-c})$ for arbitrary large $c$, where $n$ is the number of vertices. The proof relies on a new tail bound for the cumulant expansion of the Laplace transform, which is of independent interest.