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

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 $δ$) 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.

Figures (11)

• Figure 1: The solution of EHN problem for one dimension -- the alternating lines of unit length with one excluded point (at the right edge in our example).
• Figure 2: Illustration of the fact that $4\leqslant q \leqslant 7$: (a) The Moser's spindle -- four-chromatic unit distance graph on 11 vertices proving the necessity of minimum 4 colors for coloring plane such that there are no points of the same color at the unit distance apart; (b) The coloring of the plane with 7 colors such that there are no points of the same color at the unit distance apart.
• Figure 3: (a): The schematic picture of the model. A particle at the center of the ring interacts only with its ring-neighbors, which are, by definition, particles at $R \pm \delta/2$ distance from the chosen one. The dotted square represents the area outside of which the colors of particles are fixed - the implementation of fixed boundary conditions; (b): The schematic picture of energy zones. All particles have the same properties but are depicted differently depending on the zone they belong to. The inside energy is calculated only between open circled particles and each pair of them counted twice. Pairs which consist of one open circled particle from inside zone and dotted circled particle from adjoining zone counted once. They contribute to outside energy. The interaction of pairs of dotted circled particles doesn't contribute to energy. Interactions with ring symbol particles also omitted. It is worth mentioning that one should not be deluded by sparse density of particles - in reality the density is much higher and the example is given only for illustrative purposes.
• Figure 4: Typical energy minimization curves for $q=6$. (a): Greedy algorithm; (b): Simulated annealing
• Figure 5: Typical vacuum configurations. (a): $q$ = 2; (b): $q$ = 3