Lipschitz functions on weak expanders

TL;DR

This paper extends understanding of Lipschitz functions on weak expanders, showing they typically have small fluctuations under broader conditions, using advanced combinatorial and information-theoretic techniques.

Contribution

It generalizes previous results by relaxing expansion requirements and allowing larger Lipschitz constants, employing novel graph container and entropy methods.

Findings

Lipschitz functions on weak expanders have small fluctuations.

The results hold under broader expansion conditions.

Techniques combine graph container and entropy methods.

Abstract

Given a connected finite graph $G$, an integer-valued function $f$ on $V(G)$ is called $M$-Lipschitz if the value of $f$ changes by at most $M$ along the edges of $G$. In 2013, Peled, Samotij, and Yehudayoff showed that random $M$-Lipschitz functions on graphs with sufficiently good expansion typically exhibit small fluctuations, giving sharp bounds on the typical range of such functions, assuming $M$ is not too large. We prove that the same conclusion holds under a relaxed expansion condition and for larger $M$, (partially) answering questions of Peled et al. Our techniques involve a combination of Sapozhenko's graph container methods and entropy methods from information theory.