# Entropy inequalities for factors of IID

## TL;DR

This paper develops new entropy inequalities for factors of IID processes on infinite trees, providing a versatile method to derive such inequalities with applications in graph eigenvector analysis.

## Contribution

It introduces a general approach to find and prove entropy inequalities for broader classes of factor processes with fewer symmetries.

## Key findings

- New entropy inequalities for factors of IID on infinite trees
- A general 'recipe' for deriving entropy inequalities
- Application to eigenvector analysis of random regular graphs

## 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.