# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] Peter Antal and Agoston Pisztora, On the chemical distance for supercritical Bernoulli percolation, Ann. Probab. 24 (1996), no. 2, 1036 – 1048. · Zbl 0871.60089 · doi:10.1214/aop/1039639377 · doi.org [2] Eric Babson and Itai Benjamini, Cut sets and normed cohomology with applications to percolation, Proc. Amer. Math. Soc. 127 (1999), no. 2, 589 – 597. · Zbl 0910.60075 [3] Jean-Dominique Deuschel and Agoston Pisztora, Surface order large deviations for high-density percolation, Probab. Theory Related Fields 104 (1996), no. 4, 467 – 482. · Zbl 0842.60023 · doi:10.1007/BF01198162 · doi.org [4] Guy Gielis and Geoffrey Grimmett, Rigidity of the interface in percolation and random-cluster models, J. Statist. Phys. 109 (2002), no. 1-2, 1 – 37. · Zbl 1025.82007 · doi:10.1023/A:1019950525471 · doi.org [5] Alan Hammond, Greedy lattice animals: geometry and criticality, Ann. Probab. 34 (2006), no. 2, 593 – 637. · Zbl 1097.60081 · doi:10.1214/009117905000000693 · doi.org [6] Geoffrey R. Grimmett and Alexander E. Holroyd, Entanglement in percolation, Proc. London Math. Soc. (3) 81 (2000), no. 2, 485 – 512. · Zbl 1026.60109 · doi:10.1112/S0024611500012521 · doi.org [7] Harry Kesten, Aspects of first passage percolation, École d’été de probabilités de Saint-Flour, XIV — 1984, Lecture Notes in Math., vol. 1180, Springer, Berlin, 1986, pp. 125 – 264. · Zbl 0602.60098 · doi:10.1007/BFb0074919 · doi.org [8] Harry Kesten and Yu Zhang, The probability of a large finite cluster in supercritical Bernoulli percolation, Ann. Probab. 18 (1990), no. 2, 537 – 555. · Zbl 0705.60092 [9] Gábor Pete, A note on percolation on \Bbb Z^\?: isoperimetric profile via exponential cluster repulsion, Electron. Commun. Probab. 13 (2008), 377 – 392. · Zbl 1191.60116 · doi:10.1214/ECP.v13-1390 · doi.org [10] Agoston Pisztora, Surface order large deviations for Ising, Potts and percolation models, Probab. Theory Related Fields 104 (1996), no. 4, 427 – 466. · Zbl 0842.60022 · doi:10.1007/BF01198161 · doi.org [11] Ádám Timár, Cutsets in infinite graphs, Combin. Probab. Comput. 16 (2007), no. 1, 159 – 166. · Zbl 1170.05033 · doi:10.1017/S0963548306007838 · doi.org
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.