A local limit theorem for triple connections in subcritical Bernoulli percolation

Abstract

We prove a local limit theorem for the probability of a site to be connected by disjoint paths to three points in subcritical Bernoulli percolation on Z d , d ≥ 2 in the limit where their distances tend to infinity.

On the Gibbs states of the noncritical Potts model on $$\mathbb Z ^2$$

“…is strictly convex and quadratic around its minimum point; see [9,Lemma 3]. Let x be the unique minimizer of φ.…”

“…Nevertheless, the idea remains the same: The tripod has Gaussian fluctuations, therefore it intersects a small box with probability going to 0. In the case of percolation, fluctuations of tripods on the level of local limit results were studied in [9]. We are not after a full local limit picture here, and merely explain how techniques from [10] allow to derive (27).…”

“…uniformly in t ≤ Ck ε , W = {w 1 , w 2 , w 3 } ⊂ ∂Λ t and y, z ∈ Λ ε 1 k , where the sum is over C 0 compatible with S W (t, y) and where in the last line we used classical ratio-mixing properties of subcritical randomcluster measures [3, Theorem 3.4] and (6) to compare µ f D (C 0 (t, y) = C 0 ) and µ f D (C 0 (t, z) = C z−y 0 ). The function φ W has a non-degenerate quadratic minimum at some x min (W ) (see [9,Lemma 3]). In view of the homogeneity of τ , a quadratic expansion around x min yields Lemmas 5.3 and 5.4 imply that…”

Some results on the asymptotic behavior of finite connection probabilities in percolation

“…For any two distinct lattice points x, y ∈ , looking at 1 {0↔x, 0 c } and 1 {x↔y, y c } as functions of (E {x} , E c ), they are both nondecreasing on E {x} and nonincreasing on E c . Therefore, by Theorem 2.1 in [van den Berg et al 2006…”

Ornstein-Zernike behavior for Ising models with infinite-range interactions

“…Let us mention that the approach developed by Dima and his collaborators led to a great variety of applications to important problems in equilibrium statistical mechanics, related to the properties of interfaces in planar lattice systems [22,15,23,19,18,28,40,27,26], the effect of a stretching force on self-interacting polymers with or without disorder [29,30,31,33,32,34,25], the asymptotic behavior of more general correlation functions [14,9,12,10,39], etc. 1.2.…”