On Sidorenko's conjecture for determinants and Gaussian Markov random fields

TL;DR

This paper explores determinant inequalities related to Sidorenko's conjecture, establishing entropy bounds for Gaussian Markov random fields on bipartite graphs and connecting these to graph theory and zeta functions.

Contribution

It introduces a novel entropy inequality for homogeneous GMRFs on bipartite graphs and links Sidorenko's conjecture to Gaussian fields using graph limit theory.

Findings

Entropy of homogeneous GMRFs on bipartite graphs is bounded below by edge and point entropies.

The inequality extends to graphs with non-negative edge correlations.

Connections to Ihara zeta function and spanning trees are discussed.

Abstract

We study a class of determinant inequalities that are closely related to Sidorenko's famous conjecture (Also conjectured by Erd\H os and Simonovits in a different form). Our main result can also be interpreted as an entropy inequality for Gaussian Markov random fields (GMRF). We call a GMRF on a finite graph $G$ homogeneous if the marginal distributions on the edges are all identical. We show that if $G$ is bipartite then the differential entropy of any homogeneous GMRF on $G$ is at least $|E(G)|$ times the edge entropy plus $|V(G)|-2|E(G)|$ times the point entropy. We also show that in the case of non-negative correlation on edges, the result holds for an arbitrary graph $G$. The connection between Sidorenko's conjecture and GMRF's is established via a large deviation principle on high dimensional spheres combined with graph limit theory. Connection with Ihara zeta function and the number of spanning trees is also discussed.