Concentration of information on discrete groups

TL;DR

This paper studies varentropy in discrete groups, establishing a universal bound for conjugacy-invariant random walks that parallels continuous log-concave vector results, with implications for information concentration.

Contribution

It introduces an approximate tensorization inequality for varentropy on discrete groups, linking it to the free Abelian case and demonstrating a universal bound based on the number of generators.

Findings

Varentropy of conjugacy-invariant random walks is bounded by the number of generators.

Conjugacy-invariant random walks have non-negative Bakry-Émery curvature.

The results provide a discrete analogue of known continuous log-concave vector inequalities.

Abstract

Motivated by the Asymptotic Equipartition Property and its recently discovered role in the cutoff phenomenon, we initiate the systematic study of varentropy on discrete groups. Our main result is an approximate tensorization inequality which asserts that the varentropy of any conjugacy-invariant random walk is, up to a universal multiplicative constant, at most that of the free Abelian random walk with the same jump rates. In particular, it is always bounded by the number d of generators, uniformly in time and in the size of the group. This universal estimate is sharp and can be seen as a discrete analogue of a celebrated result of Bobkov and Madiman concerning random d-dimensional vectors with a log-concave density (AOP 2011). A key ingredient in our proof is the fact that conjugacy-invariant random walks have non-negative Bakry-\'Emery curvature, a result which seems new and of independent interest.