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

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 results can also be interpreted as entropy inequalities 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 $G$ satisfies Sidorenko's conjecture 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 prove this inequality in a large class of graphs for which Sidorenko's conjecture is not verified including the so-called M\"obius ladder: $K_{5,5}\setminus C_{10}$. 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.