Critical Percolation and the Minimal Spanning Tree in Slabs

Abstract

The minimal spanning forest on ℤd is known to consist of a single tree for d ≤ 2 and is conjectured to consist of infinitely many trees for large d. In this paper, we prove that there is a single tree for quasi‐planar graphs such as ℤ2 × {0,…,k}d−2. Our method relies on generalizations of the “gluing lemma” of Duminil‐Copin, Sidoravicius, and Tassion. A related result is that critical Bernoulli percolation on a slab satisfies the box‐crossing property. Its proof is based on a new Russo‐Seymour‐Welsh‐type theorem for quasi‐planar graphs. Thus, at criticality, the probability of an open path from 0 of diameter n decays polynomially in n. This strengthens the result of Duminil‐Copin et al., where the absence of an infinite cluster at criticality was first established. © 2017 Wiley Periodicals, Inc.

Geometry of the minimal spanning tree of a random 3-regular graph

“…Studies related to the MST in several other directions were undertaken in [18,28,77,78,80]. An account of certain structural and connectivity properties of minimal spanning forests can be found in [14,16,67,76] and the references therein. For an account of the scaling limit of minimal spanning trees in subsets of Z 2 with respect to the topology introduced by Aizenman, Burchard, Newman, and Wilson, see, e.g., [7,49].…”

Ground state stability in two spin glass models

“…The crossover dimension was found to be six in [11]. It should be noted that this result, related to the minimal spanning tree, is rigorous only in dimension two (or in quasi-planar lattices [18]).…”

A Central Limit Theorem for First Passage Percolation in the Slab

“…Two main ingredients for our proof are an adaptation of the martingale central limit theorem in [7] and the Russo Seymour Welsh Theorem for Bernouilli percolation on slabs [8]. The analogous result on Z 2 was proved by Kesten and Zhang [3] by adapting the martingale central limit theorem and the RSW Theorem on Z 2 , and considering the passage time within annuli A n,2n .…”

“…Applying the RSW Theorem proved in [8], we see that with probability uniformly bounded from below, there is a p c -closed circuit in A(2 p , 2 p+1 ). Still denote by C p the innermost such circuit (given a fixed ordering of edges).…”

“…To deal with this, we modify the Def. (8) as The increments ( p ) p≤q will be different from the (C p ) p≤q , but only by a finite number (in fact it is bounded by C k , where k is the width of the slab). Therefore it still satisfies the condition of McLeish's CLT [7] and an application of that CLT concludes Theorem 1.…”