an:06802188
Zbl 1380.82025
Newman, Charles; Tassion, Vincent; Wu, Wei
Critical percolation and the minimal spanning tree in slabs
EN
Commun. Pure Appl. Math. 70, No. 11, 2084-2120 (2017).
00370886
2017
j
82B43 05C05 82B27
percolation; minimal spanning tree; critical Bernoulli percolation
Authors' abstract: The minimal spanning forest on \(\mathbb Z^d\) is known to consist of a single tree for \(d\leq 2\) 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 \(\mathbb Z^2\times\{0,\dots,k\}^{(d-2)}\). Our method relies on generalizations of the ``gluing lemma'' of \textit{H. Duminil-Copin} et al. [Commun. Pure Appl. Math. 69, No. 7, 1397--1411 (2016; Zbl 1342.82076)]. 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. [loc. cit.] where the absence of an infinite cluster at criticality was first established.
E. Ahmed (Mansoura)
Zbl 1342.82076