Mutual information decay for factors of i.i.d.

Abstract

This paper is concerned with factors of independent and identically distributed processes on the $d$ -regular tree for $d\geq 3$ . We study the mutual information of values on two given vertices. If the vertices are neighbors (i.e. their distance is $1$ ), then a known inequality between the entropy of a vertex and the entropy of an edge provides an upper bound for the (normalized) mutual information. In this paper we obtain upper bounds for vertices at an arbitrary distance $k$ , of order $(d-1)^{-k/2}$ . Although these bounds are sharp, we also show that an interesting phenomenon occurs here: for any fixed process, the rate of mutual information decay is much faster, essentially of order $(d-1)^{-k}$ .

Entropy and expansion

“…When Γ is a free group and the weights are all equal, we get back an inequality that was known by Bowen [10] as the fact the so-called f -invariant is nonnegative for factors of the Bernoulli shift. See also [16,Theorem 2.3], where this inequality is explicitly stated.…”

“…It is easy to construct factor-of-IID processes for which the ratio H(X U 0 )/H(X v ) tends to the above coefficient, see the example described in [16,Section 5.1], showing that this coefficient is indeed the best possible.…”

“…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].…”

Uniform even subgraphs and graphical representations of Ising as factors of i.i.d

“…Although there has been a lot of progress in the case of amenable graphs [2,1,25,46,47,44,45,43], the nonamenable setting remains a rather unexplored territory; see Bowen [13] for a general result in this direction and Lyons [35] for a survey of results on a tree. See also [37,8,21,7,14,27,40] for other relevant results. We elaborate a bit on this now and explain how our results have some consequences for the gradient of Ising model on planar graphs.…”