High-Dimensional Lipschitz Functions are Typically Flat

TL;DR

This paper demonstrates that in high dimensions, typical Lipschitz and homomorphism height functions on tori are extremely flat, with bounded variance and limited value ranges, revealing a phase transition in their roughness.

Contribution

It establishes the flatness of high-dimensional Lipschitz functions, extends results to enhanced tori, and analyzes the behavior of proper 3-colorings, addressing conjectures and questions in the field.

Findings

Typical functions are very flat with bounded variance.

Functions take at most $( ext{log } n)^{1/d}$ values in high dimensions.

Proper 3-colorings are nearly constant on even or odd sub-lattices.

Abstract

A homomorphism height function on the $d$-dimensional torus $\mathbb{Z}_n^d$ is a function taking integer values on the vertices of the torus with consecutive integers assigned to adjacent vertices. A Lipschitz height function is defined similarly but may also take equal values on adjacent vertices. In each model, we consider the uniform distribution over such functions, subject to boundary conditions. We prove that in high dimensions, with zero boundary values, a typical function is very flat, having bounded variance at any fixed vertex and taking at most $C(\log n)^{1/d}$ values with high probability. Our results extend to any dimension $d\ge 2$, if $\mathbb{Z}_n^d$ is replaced by an enhanced version of it, the torus $\mathbb{Z}_n^d\times\mathbb{Z}_2^{d_0}$ for some fixed $d_0$. This establishes one side of a conjectured roughening transition in $2$ dimensions. The full transition is established for a class of tori with non-equal side lengths. We also find that when $d$ is taken to infinity while $n$ remains fixed, a typical function takes at most $r$ values with high probability, where $r=5$ for the homomorphism model and $r=4$ for the Lipschitz model. Suitable generalizations are obtained when $n$ grows with $d$. Our results apply also to the related model of uniform 3-coloring and establish, for certain boundary conditions, that a uniformly sampled proper 3-coloring of $\mathbb{Z}_n^d$ will be nearly constant on either the even or odd sub-lattice. Our proofs are based on a combinatorial transformation and on a careful analysis of the properties of a class of cutsets which we term odd cutsets. For the Lipschitz model, our results rely also on a bijection of Yadin. This work generalizes results of Galvin and Kahn, refutes a conjecture of Benjamini, Yadin and Yehudayoff and answers a question of Benjamini, H\"aggstr\"om and Mossel.