Percolation Properties of the Non-ideal Gas

Abstract

We estimate locations of the regions of the percolation and of the non-percolation in the plane (λ, β): the Poisson rate -the inverse temperature, for interacted particle systems in finite dimension Euclidean spaces. Our results about the percolation and about the nonpercolation are obtained under different assumptions. The intersection of two groups of the assumptions reduces the results to two dimension Euclidean space, R 2 , and to a potential function of the interactions having a hard core.The technics for the percolation proof is based on a contour method which is applied to a discretization of the Euclidean space. The technics for the non-percolation proof is based on the coupling of the Gibbs field with a branching process.

Continuum percolation for Gibbs point processes

“…for a Strauss hard core process (ϕ(x, y) = ½{ x − y ≥ r}), one would expect a lower bound on the percolation radius of r/2. The main difference is however, that the percolation result of [13] is valid only in two dimension, whereas our result holds in any dimension d ≥ 2.…”

“…In [13] there are various assumptions on ϕ including our condition (i) of Proposition 4.1 and an attraction condition. Namely, ϕ is said to have an attractive tail if there exists two constants r a < r ′ a such that ϕ(x, y) > 1, whenever r a ≤ x − y ≤ r ′ a .…”

“…ϕ(x, y) = 0 whenever x − y ≤ r, and ϕ(x, y) → 1 fast enough as x − y → ∞. The non-percolation result in [13] treats only the hard core case. However, unless for the percolation result, the authors proof is quite different from ours.…”

“…The main idea for the proof of percolation is based on techniques close to the contour method in lattice models. In particular, it is a modification of the arguments in [13] and [1]. Let condition (P) be satisfied with constants r, δ > 0, and let R > r. Choose a m ∈ N such that m > √ d/(R − r).…”

“…The aim of this work is to extend the results of [13,1] to any dimension d ≥ 2 and to very general Gibbs point processes. We give a percolation condition on the conditional intensity of a Gibbs process, which is easily understandable and which is satisfied for most Gibbs processes, provided the percolation radius R is large enough.…”