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arXiv · 1801.08425

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

Péter Csikvári·Balázs Szegedy
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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.

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Péter Csikvári, Balázs Szegedy. 2018-01-25. On Sidorenko's conjecture for determinants and Gaussian Markov random fields. https://arxiv.org/abs/1801.08425

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