# Mutual information decay for factors of IID

## TL;DR

This paper investigates how mutual information between vertices in factor of i.i.d. processes on regular trees decays with distance, revealing a faster decay rate than previously established bounds.

## Contribution

It provides new upper bounds for mutual information decay at arbitrary distances and uncovers a faster decay rate for fixed processes compared to general bounds.

## Key findings

- Mutual information decays as (d-1)^{-k/2} for vertices at distance k.
- For fixed processes, the decay rate is essentially (d-1)^{-k}.
- Bounds are sharp, indicating precise characterization of decay rates.

## Abstract

This paper is concerned with factor of i.i.d. processes on the $d$-regular tree for $d \geq 3$. 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 $(d-1)^{-k}$.