Non-local Potts model on random lattice and chromatic number of a plane

TL;DR

This paper explores a non-local Potts model on a random lattice, analyzing its vacuum states and patterns, and discusses its potential connection to the chromatic number of a plane problem.

Contribution

It introduces a non-local q-color Potts model on a random lattice and investigates its vacuum states through numerical simulations, linking it to the chromatic number of a plane.

Findings

Numerical simulations reveal distinct pattern formations in the model.

The model's behavior suggests a relation to the chromatic number of a plane.

Qualitative features of the vacuum states are characterized.

Abstract

Statistical models are widely used for the investigation of complex system's behavior. Most of the models considered in the literature are formulated on regular lattices with nearest-neighbor interactions. The models with non-local interaction kernels have been less studied. In this article, we investigate an example of such a model - the non-local q-color Potts model on a random d=2 lattice. Only the same color spins at a unit distance (within some small margin $\delta$) interact. We study the vacuum states of this model and present the results of numerical simulations and discuss qualitative features of the corresponding patterns. Conjectured relation with the chromatic number of a plane problem is discussed.