Boundary-connectivity via Graph Theory

We generalize theorems of Kesten and Deuschel-Pisztora about the connectedness of the exterior boundary of a connected subset of Zd, where “connectedness” and “boundary” are understood with respect to various graphs on the vertices of Zd. 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. Denote by Z the usual nearest-neighbor lattice on Z, i.e., two points of Z are adjacent if they differ only in one coordinate, by 1. Let Zd∗ be the graph on the same vertex set and edges between every two distinct points that differ in every coordinate by at most 1. We say that a set of vertices in Z is *-connected if it is connected in the graph Zd∗. In [DP] Deuschel and Pisztora prove that the part of the outer vertex boundary of a finite connected subgraph C in Zd∗ that is visible from infinity (the exterior boundary) is *-connected. Earlier, Kesten [K] showed that the set of points in the *-boundary of a connected subgraph C ⊂ Zd∗ that are Z-visible from infinity is connected in Z. Similar results were proved about the case when C is in an n× n box of Z [DP], or Zd∗ [H]. See the second paragraphs of Theorem 3 and Theorem 4 for the precise statements. We generalize these results about Z and Zd∗ to a very general family of pairs of graphs; see Lemma 2, Theorem 3 and Theorem 4. Our method also gives an elementary and short alternative to the original proofs for the cubic grid case. This approach seems to be efficient to treat possible other questions about the connectedness of boundaries. Although [K] mentions that some use of algebraic topology seems to be unavoidable, the greater generality (and simplicity) of our proof is a result of using purely graph-theoretic arguments. Also, it makes slight modifications of the results (such as considering boundaries in some subset of Z instead of boundaries in Z) straightforward, while previously one had to go through the original proofs and make significant modifications. In two dimensions, the use of some duality argument makes connectedness of boundaries more straightforward to prove. The lack of duality (that is, the correspondance that a cycle in one graph is a separating set in its dual) in higher Received by the editors March 25, 2010 and, in revised form, February 21, 2011; July 1, 2011; and July 5, 2011. 2010 Mathematics Subject Classification. Primary 05C10, 05C63; Secondary 20F65, 60K35. c ©2012 American Mathematical Society Reverts to public domain 28 years from publication