Random Lipschitz functions on graphs with weak expansion

TL;DR

This paper extends previous results on the behavior of random Lipschitz functions on graphs with weak expansion, exploring their range and phase transitions in more general settings.

Contribution

It generalizes earlier findings to random M-Lipschitz and real-valued Lipschitz functions, analyzing their properties on graphs with weak expansion.

Findings

Super-constant range for certain graphs with weak expansion

Sharp phase transition identified around specific parameters

Extension of results to real-valued Lipschitz functions

Abstract

Benjamini, Yadin, and Yehudayoff (2007) showed that if the maximum degree of a graph $G$ is 'sub-logarithmic,' then the typical range of random $\mathbb Z$-homomorphisms is super-constant. Furthermore, they showed that there is a sharp transition on the range of random $\mathbb Z$-homomorphisms on the graph $C_{n,k}$, the tensor product of the $n$-cycle and the complete graph on $k$ vertices with self-loops, around $k=2\log n$. We extend (to some extent) their results to random $M$-Lipschitz functions and random real-valued Lipschitz functions.