Grounded Lipschitz functions on trees are typically flat

TL;DR

This paper investigates grounded Lipschitz functions on trees, showing that the root's value is typically very close to zero with high probability, indicating such functions are usually flat.

Contribution

It establishes that the probability of the root's value deviating significantly from zero is doubly-exponentially small, providing new bounds for both discrete and continuous grounded Lipschitz functions.

Findings

Root value deviations are doubly-exponentially unlikely

Grounded Lipschitz functions are typically flat

Similar bounds hold for both integer-valued and real-valued functions

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 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.