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 Erdős 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 Möbius ladder: K 5,5 \ 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.