Entropy inequalities for factors of IID

Abstract

This paper is concerned with certain invariant random processes (called factors of IID) on infinite trees. Given such a process, one can assign entropies to different finite subgraphs of the tree. There are linear inequalities between these entropies that hold for any factor of IID process (e.g. "edge versus vertex" or "star versus edge"). These inequalities turned out to be very useful: they have several applications already, the most recent one is the Backhausz-Szegedy result on the eigenvectors of random regular graphs.We present new entropy inequalities in this paper. In fact, our approach provides a general "recipe" for how to find and prove such inequalities. Our key tool is a generalization of the edge-vertex inequality for a broader class of factor processes with fewer symmetries.2010 Mathematics Subject Classification. 37A35, 60K35, 37A50, 05E18.

Local Problems on Trees from the Perspectives of Distributed Algorithms, Finitary Factors, and Descriptive Combinatorics

“…We note that getting only "approximate" solutions, that is, solutions where a small fraction of vertices does not have to satisfy the constraints of a given problem, is intrinsic in this correspondence. Regardless, there are many techniques, such as entropy inequality [BGH19] or correlation decay [BSV15], and particular results such as the aforementioned perfect matching problem [LN11] that provide lower and upper bounds, respectively, in our setting as well. We refer the reader to [Lyo17,BGHV18] for a comprehensive summary of the field.…”

“…In this paper, we describe examples of graphs of chromatic number up to 2∆ − 2 such that the corresponding homomorphism problem is not in BOREL, see Section 4. The theory of entropy inequalities see [BGH19] implies that if G is a cycle on more than 9 vertices, then Π G ∈ fiid.…”

Entropy and expansion

“…the graph distance in G. Block factors are also called finite-radius factors. For a more detailed introduction to factors of IID, see [3,Section 2].…”

“…All previous proofs of (2) are based on counting arguments for random regular graphs or for random permutations, and as such they heavily build on the acyclic nature of T d , see [4,10,25]. Further inequalities and generalizations can be found in [16,3].…”