Grounded Lipschitz functions on trees are typically flat

Abstract

A grounded $M$-Lipschitz function on a rooted $d$-ary tree is an integer valued map on the vertices that changes by at most $M$ along edges and attains the value zero on the leaves. We study the behavior of such functions, specifically, their typical value at the root $v_0$ of the tree. We prove that the probability that the value of a uniformly chosen random function at $v_0$ is more than $M+t$ is doubly-exponentially small in $t$. We also show a similar bound for continuous (real-valued) grounded Lipschitz functions.