Isaev, MikhailMcKay, Brendan D.Zhang, Rui Ray2025-05-232025-05-230095-8956ORCID:/0000-0002-3553-0496/work/184098239http://www.scopus.com/inward/record.url?scp=85215827671&partnerID=8YFLogxKhttps://hdl.handle.net/1885/733752017An 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.Supported by Australian Research Council grant DP250101611.52en© 2025 The Author(s)CumulantEulerian orientationGraphIce-modelSpanning tree202510.1016/j.jctb.2025.01.00285215827671