Cumulant expansion for counting Eulerian orientations

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 log8 n , we derive an asymptotic expansion for this count that approximates it to precision O ( n - c ) for arbitrarily 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. (c) 2025 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).