Lipschitz Functions on Expanders are Typically Flat

TL;DR

This paper demonstrates that random Lipschitz functions on expander graphs tend to be nearly constant, with most functions exhibiting minimal variation and taking very few values, revealing a typical flatness behavior.

Contribution

It provides the first rigorous analysis showing that Lipschitz functions on expanders are usually flat, with most functions taking very limited values and exhibiting small fluctuations.

Findings

Most Lipschitz functions on expanders take only M+1 values.

Probability of functions taking more values decays double exponentially.

Functions exhibit very small fluctuations on typical expanders.

Abstract

This work studies the typical behavior of random integer-valued Lipschitz functions on expander graphs with sufficiently good expansion. We consider two families of functions: M-Lipschitz functions (functions that change by at most M along edges) and integer-homomorphisms (functions that change by exactly 1 along edges). We prove that such functions typically exhibit very small fluctuations. For instance, we show that a uniformly chosen M-Lipschitz function takes only M+1 values on most of the graph, with a double exponential decay for the probability to take other values.