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Critical percolation and the minimal spanning tree in slabs. (English) Zbl 1380.82025
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 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.

82B43 Percolation
05C05 Trees
82B27 Critical phenomena in equilibrium statistical mechanics
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