On the local convergence of integer-valued Lipschitz functions on regular trees

TL;DR

This paper investigates the local convergence properties of integer-valued Lipschitz functions on regular trees, revealing a phase transition at degree 8 and providing new insights into the structure of Gibbs measures and localization phenomena.

Contribution

It establishes the existence or non-existence of weak limits for 1-Lipschitz functions on regular trees depending on degree, and introduces a fixed point convergence approach for analyzing these functions.

Findings

Weak limit exists for degrees 2 to 7, but not for degrees 8 and above.

Root value exhibits an even-odd oscillation pattern for large degrees, affecting convergence.

Provides an alternative proof of localization and results on Lipschitz functions on non-amenable graphs.

Abstract

We study random integer-valued Lipschitz functions on regular trees. It was shown by Peled, Samotij and Yehudayoff that such functions are localized, however, finer questions about the structure of Gibbs measures remain unanswered. Our main result is that the weak limit of a uniformly chosen 1-Lipschitz function with 0 boundary condition on a $d$-ary tree of height $n$ exists as $n \to \infty$ if $2 \le d \le 7$, but not if $d \ge 8$, thereby partially answering a question posed by Peled, Samotij and Yehudayoff. For large $d$, the value at the root alternates between being almost entirely concentrated on 0 for even $n$ and being roughly uniform on $\{-1,0,1\}$ for odd $n$, leading to different limits as $n$ approaches infinity along evens or odds. For $d \ge 8$, the essence of this phenomenon is preserved, which obstructs the convergence. For $d \le 7$, this phenomenon ceases to exist, and the law of the value at the root loses its connection with the parity of $n$. Along the way, we also obtain an alternative proof of localization. The key idea is a fixed point convergence result for a related operator on $\ell^\infty$, and a procedure to show that the iterations get into a `basin of attraction' of the fixed point. We also prove some accompanying analogous `even-odd phenomenon' type results about $M$-lipschitz functions on general non-amenable graphs with high enough expansion (this includes for example the large $d$ case for regular trees). We also prove a convergence result for 1-Lipschitz functions with $\{0,1\}$ boundary condition. This last result relies on an absolute value FKG for uniform 1-Lipschitz functions when shifted by $1/2$.