  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.
