# Entropy inequalities for factors of IID

### Ágnes Backhausz

Eötvös Loránd University and Alfréd Rényi Institute of Mathematics, Budapest, Hungary### Balázs Gerencsér

Alfréd Rényi Institute of Mathematics and Eötvös Loránd University, Budapest, Hungary### Viktor Harangi

Alfréd Rényi Institute of Mathematics, Budapest, Hungary

## 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.

## Cite this article

Ágnes Backhausz, Balázs Gerencsér, Viktor Harangi, Entropy inequalities for factors of IID. Groups Geom. Dyn. 13 (2019), no. 2, pp. 389–414

DOI 10.4171/GGD/492