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Boundary-connectivity via graph theory. (English) Zbl 1259.05049
Summary: We generalize theorems of H. Kesten [Lect. Notes Math. 1180, 125–264 (1986; Zbl 0602.60098)] and of J. Deuschel and A. Pisztora [Probab. Theory Relat. Fields 104, No. 4, 467–482 (1996; Zbl 0842.60023)] about the connectedness of the exterior boundary of a connected subset of \({\mathbb{Z}}^d\), where “connectedness” and “boundary” are understood with respect to various graphs on the vertices of \({\mathbb{Z}}^d\). These theorems are widely used in statistical physics and related areas of probability. We provide simple and elementary proofs of their results. It turns out that the proper way of viewing these questions is graph theory instead of topology.

MSC:
05C10 Planar graphs; geometric and topological aspects of graph theory
05C40 Connectivity
05C63 Infinite graphs
20F65 Geometric group theory
60K35 Interacting random processes; statistical mechanics type models; percolation theory
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