Identities and Inequalities for Tree Entropy

TL;DR

This paper explores new mathematical expressions for tree entropy, demonstrating its properties like non-negativity and stochastic domination, and proves Lueck's Determinant Conjecture for Cayley-graph Laplacians using operator theory.

Contribution

It introduces novel formulas for tree entropy and establishes key properties, including non-negativity and stochastic domination, with implications for Lueck's Determinant Conjecture.

Findings

Tree entropy can be expressed via Fuglede-Kadison determinants and effective resistance.

Tree entropy is non-negative in unweighted graphs.

Tree entropy respects stochastic domination.

Abstract

The notion of tree entropy was introduced by the author as a normalized limit of the number of spanning trees in finite graphs, but is defined on random infinite rooted graphs. We give some new expressions for tree entropy; one uses Fuglede-Kadison determinants, while another uses effective resistance. We use the latter to prove that tree entropy respects stochastic domination. We also prove that tree entropy is non-negative in the unweighted case, a special case of which establishes Lueck's Determinant Conjecture for Cayley-graph Laplacians. We use techniques from the theory of operators affiliated to von Neumann algebras.