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

TL;DR

This paper explores the strong correlation between residual entropy and spanning tree entropy in ice-type models on graphs, proposing a new heuristic for estimating residual entropy and analyzing properties of random graphs.

Contribution

It reveals a strong correlation between two entropies in ice-type models and introduces a novel heuristic for residual entropy estimation in regular graphs.

Findings

Strong correlation between residual and spanning tree entropies

Proposed heuristic outperforms previous methods

Analyzed expansion and residual entropy of random graphs

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.