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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.

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