Mutual information decay for factors of iid

Gerencsér, Balázs and Harangi, Viktor (2018) Mutual information decay for factors of iid. ERGODIC THEORY AND DYNAMICAL SYSTEMS. ISSN 0143-3857

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Abstract

This paper is concerned with factor of i.i.d. processes on the d-regular tree for d≥3. We study the mutual information of the 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 decay of the mutual information is much faster, essentially of order (d−1)−k.

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