Properties of Random Connected Edge Sets in Hypercubic Lattices

Abstract

For a > 0, d = 2, 3, ..., and l = 0, 1, ..., we introduce a probability measure μₐ on a probability space Ω(l) consisting of connected edge sets in a d-dimensional hypercubic lattice, inspired by the two-point correlation function of the Ising model. We introduce a*ᵈ, the supremum of a such that μₐ is a well-defined probability measure. We show that a*ᵈ is independent of l. By means of bounding the number of elements in Ω from above, we establish the lower bound a*ᵈ > 1⁄(2d−1), and by means of constructing elements of Ω, we establish the upper bound a*ᵈ ≤ 1⁄d. The parameter a in our model plays a similar role as to that of tanh(β) in the two-point correlation function of the Ising model. As the Ising model undergoes a phase transition at the critical inverse temperature βᶜᵈ, we compare our bounds on a*ᵈ to tanh(βᶜᵈ), using values and approximations of βᶜᵈ found in literature. We find that tanh(βᶜᵈ) lies within our bounds on a*ᵈ. We relate the edge sets in our probability space Ω to the graph theoretic concepts of walks, trails, and paths, and bound the number of edge sets of length k in Ω in terms of number of such constructs of length k. In order to visualize the distribution, we employ the Metropolis-Hastings algorithm to simulate from μₐ in two dimensions constricted to a finite lattice.