Correlation between residual entropy and spanning tree entropy of ice-type models on graphs

TL;DR

This work studies the residual entropy $\rho(G)=\frac{1}{n}\log EO(G)$ of ice-type graphs and its strong, non-deterministic correlation with the spanning-tree entropy $\tau(G)$. It develops sharp bounds on $EO(G)$, proves concentration results for random graphs with given degrees, and introduces a new heuristic $\rho_\tau(G)=\widehat{\rho}(G)+\frac{1}{2}\tau_d-\frac{1}{2}\tau(G)$ that integrates spanning-tree information to predict $\rho(G)$ more accurately. The paper also presents numerical methods via Eulerian partitions to estimate $\rho(G)$ for large graphs, and analyzes Cartesian products (notably cycles and cycles-of-cliques) to illustrate the interplay between residual entropy and tree entropy across graph families. Collectively, these results offer improved estimators for $\rho(G)$, expose the role of expansion and degree structure, and provide detailed case studies (cycles, cycle-cliques, and higher-dimensional lattices) that bridge combinatorial graph theory and ice-type statistical physics.

Abstract

The logarithm of the number of Eulerian orientations, normalised by the number of vertices, is known as the residual entropy in studies of ice-type models on graphs. The spanning tree entropy depends similarly on the number of spanning trees. We demonstrate and investigate a remarkably strong, though non-deterministic, correlation between these two entropies. This leads us to propose a new heuristic estimate for the residual entropy of regular graphs that performs much better than previous heuristics. We also study the expansion properties and residual entropy of random graphs with given degrees.

Figures (7)

• Figure 1: $\rho(G)$ (vertical) versus $\tau(G)$ (horizontal) for some randomised graphs. The solid line is $\rho_\tau(G)$.
• Figure 2: Two 4-regular graphs with the same eigenvalues and number of spanning trees but different numbers of Eulerian orientations
• Figure 3: An Eulerian orientation and one of its associated Eulerian partitions.
• Figure 4: Distribution of $\lvert P\rvert$ for two 4-regular graphs on 256 vertices.
• Figure 5: Exact and estimated residual entropies for tubes $C_m\mathbin{\hbox{$\square$}} C_\infty$

Theorems & Definitions (44)

• Conjecture 2.1: schrijver1983bounds
• Conjecture 2.2
• Lemma 2.3
• Theorem 2.4: Isaev, McKay, Zhang IMZ
• Corollary 2.5
• Theorem 2.6
• Corollary 2.7
• Theorem 2.8
• Theorem 2.9
• proof