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The Growth Constant of Odd Cutsets in High Dimensions

Part of: Special processes Equilibrium statistical mechanics Graph theory

Published online by Cambridge University Press:  14 August 2017

OHAD NOY FELDHEIM  and
YINON SPINKA
OHAD NOY FELDHEIM
Affiliation:
Department of Mathematics, Stanford University, Stanford, CA 94305, USA (e-mail: [email protected])
YINON SPINKA
Affiliation:
School of Mathematical Sciences, Tel Aviv University, Tel Aviv, 69978, Israel (e-mail: [email protected])
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Abstract

A cutset is a non-empty finite subset of ℤd which is both connected and co-connected. A cutset is odd if its vertex boundary lies in the odd bipartition class of ℤd. Peled [18] suggested that the number of odd cutsets which contain the origin and have n boundary edges may be of order eΘ(n/d) as d → ∞, much smaller than the number of general cutsets, which was shown by Lebowitz and Mazel [15] to be of order dΘ(n/d). In this paper, we verify this by showing that the number of such odd cutsets is (2+o(1))n/2d.

MSC classification

Primary: 05C10: Planar graphs; geometric and topological aspects of graph theory
Secondary: 05C30: Enumeration in graph theory 82B41: Random walks, random surfaces, lattice animals, etc. 60K35: Interacting random processes; statistical mechanics type models; percolation theory 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 27 , Issue 2 , March 2018 , pp. 208 - 227
Copyright
Copyright © Cambridge University Press 2017 

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