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

Abstract

AbstractWe 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 main result can also be interpreted as an entropy inequality for Gaussian Markov random fields (GMRF). We call a GMRF on a finite graph homogeneous if the marginal distributions on the edges are all identical. We show that if is bipartite, then the differential entropy of any homogeneous GMRF on is at least times the edge entropy plus times the point entropy. We also show that in the case of non‐negative correlation on edges, the result holds for an arbitrary graph . 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. It is also observed that the system we study exhibits a phase transition on large girth regular graphs. Connection with Ihara zeta function and the number of spanning trees is also discussed.