Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Rigidity of 3-colorings of the discrete torus

Ohad Noy Feldheim and Ron Peled

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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.


Nous montrons qu’un 3-coloriage aléatoire du tore $d$-dimensionnel a une structure rigide quand la dimension $d$ est suffisamment grande. Nous montrons qu’avec grande probabilité, le coloriage prend au plus une couleur sur, soit presque tout le sous-tore pair, soit presque tout le sous-tore impair. En particulier, une couleur apparaît sur presque la moitié des sites du tore. Ce modèle correspond au cas de température 0 du modèle de Potts anti-ferromagnétique à trois états, en physique statistique.

Notre travail étend les résultats obtenus pour le tore discret avec des conditions frontière spécifiques. La difficulté principale dans cette extension est de surmonter les obstructions topologiques qui apparaissent quand aucune condition frontière n’est imposée. Localement, un 3-coloriage propre définit le gradient discret d’une fonction de hauteur à valeurs entières qui diffère exactement d’un entre deux points voisins. Néanmoins, ces fonctions locales ne définissent pas forcément une fonction de hauteur sur le tore entier, car les gradients peuvent s’accumuler en une quantité non-nulle en s’enroulant autour du tore. Notre résultat principal est qu’en grande dimension, une fonction de hauteur est bien définie sur le tore entier avec grande probabilité, ce qui permet de déduire la structure rigide du coloriage en utilisant des résultats précédents. De plus, la probabilité que le gradient s’accumule en un vecteur $m$ correspondant à l’enroulement le long de chacune des $d$ directions est exponentiellement petit en le produit de $\|m\|_{\infty}$ et de l’aire de la section du tore dans cette direction.

Au long de la preuve, nous introduisons des analogues discrets de notions de topologie algébrique. Cette théorie est développée dans une généralité qui peut permettre une utilisation pour d’autres modèles.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 54, Number 2 (2018), 952-994.

Received: 27 May 2015
Revised: 14 December 2016
Accepted: 8 March 2017
First available in Project Euclid: 25 April 2018

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 82B26: Phase transitions (general)
Secondary: 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41] 60C05: Combinatorial probability 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 05A16: Asymptotic enumeration

3-colorings Potts model Rigidity Discrete Cohomology 3-states Discrete topology


Feldheim, Ohad Noy; Peled, Ron. Rigidity of 3-colorings of the discrete torus. Ann. Inst. H. Poincaré Probab. Statist. 54 (2018), no. 2, 952--994. doi:10.1214/17-AIHP828.

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