Δ-wavy probability distributions and Potts model

Abstract

We define the wavy probability distributions on a subset and Δ-wavy probability distributions - two generalizations of the wavy probability distributions. A classification on the Δ-waviness is given. For the Δ-wavy probability distributions having normalization constant, we give a formula for this constant, to compute this constant. We show that the Potts model is a Δ-wavy probability distribution, where Δ is a partition which will be specified. For the normalization constant of Potts model, we give formulas and bounds. As to the formulas for this constant, we give two general formulas, one of them is simple while the other is more complicated, and based on independent sets, a formula for the Potts model on connected separable graphs - closed-form expressions are then obtained in several cases -, and a formula for the Potts model on graphs with a vertex of degree 2 - a recurrence relation is then obtained for the normalization constant of Potts model on Cn; the cycle graph with n vertices; the normalization constant of Ising model on Cn is computed using this relation. As to the bounds for the normalization constant, we present two ways to obtain such bounds; we illustrate these ways giving a general lower bound, and a lower bound and an upper one when the model is the Potts model on Gn,n, the square grid graph, n = 6k, k ≥ 1 - two upper bounds for the free energy per site of this model are then obtained, one of them being in the limit. A sampling method for the Δ-wavy probability distributions is given and, as a result, a sampling method for the Potts is given. This method - that for the Potts model too - has two steps, Step 1 and Step 2, when |Δ| > 1 and one step, Step 2 only, when |Δ| = 1. For the Potts model, Step 1 is, in general, difficult. As to Step 2, for the Potts model too, using the Gibbs sampler in a generalized sense, we obtain an exact (not approximate) sampling method having p + 1 steps (p + 1 substeps of Step 2), where p = |I|; I is an independent set, best, a maximum independent set, best, a maximum independent set - for the Potts model on Gn1,n2,…,nd , the d-dimensional grid graph, d ≥ 1, n1, n2,…, nd ≥ 1, n1n2·…·nd ≥ 2; we obtain an exact sampling method for half or half+1 vertices.