Rigidity of 3-colorings of the discrete torus

TL;DR

This paper proves that in high dimensions, a random proper 3-coloring of a discrete torus exhibits a highly rigid structure, with one color dominating almost all sites on either the even or odd sub-torus, extending previous results to models without boundary conditions.

Contribution

It introduces a new approach to analyze 3-colorings on tori without boundary conditions, establishing the high-probability existence of a global height function and quantifying the probability of winding configurations.

Findings

Colorings are nearly monochromatic on large sub-tori.

Probability of non-zero winding is exponentially small.

Global height functions exist with high probability in high dimensions.

Abstract

We prove that a uniformly chosen proper $3$-coloring of the $d$-dimensional discrete torus has a very rigid structure when the dimension $d$ is sufficiently high. We show that with high probability the coloring takes just one color on almost all of either the even or the odd sub-torus. In particular, one color appears on nearly half of the torus sites. This model is the zero temperature case of the $3$-state anti-ferromagnetic Potts model from statistical physics. Our work extends previously obtained results for the discrete torus with specific boundary conditions. The main challenge in this extension is to overcome certain topological obstructions which appear when no boundary conditions are imposed on the model. Locally, a proper $3$-coloring defines the discrete gradient of an integer-valued height function which changes by exactly one between adjacent sites. However, these locally-defined functions do not always yield a height function on the entire torus, as the gradients may accumulate to a non-zero quantity when winding around the torus. Our main result is that in high dimensions, a global height function is well defined with high probability, allowing to deduce the rigid structure of the coloring from previously known results. Moreover, the probability that the gradients accumulate to a vector $m$, corresponding to the winding in each of the $d$ directions, is at most exponentially small in the product of $\|m\|_\infty$ and the area of a cross-section of the torus. In the course of the proof we develop discrete analogues of notions from algebraic topology. This theory is developed in some generality and may be of use in the study of other models.