Hostname: page-component-77f85d65b8-g98kq Total loading time: 0 Render date: 2026-04-21T14:23:27.074Z Has data issue: false hasContentIssue false
  • English
  • Français

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: ohadf@netvision.net.il)
YINON SPINKA
Affiliation:
School of Mathematical Sciences, Tel Aviv University, Tel Aviv, 69978, Israel (e-mail: yinonspi@post.tau.ac.il)
Get access Rights & Permissions [Opens in a new window]

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 e Θ(n/d) as d → ∞, much smaller than the number of general cutsets, which was shown by Lebowitz and Mazel [15] to be of order d Θ(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

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 27 , Issue 2 , March 2018 , pp. 208 - 227
Copyright
Copyright © Cambridge University Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1] Balister, P. and Bollobás, B. (2007) Counting regions with bounded surface area. Commun. Math. Phys. 273 305–315.Google Scholar
[2] Bollobás, B. (2006) The Art of Mathematics: Coffee Time in Memphis, Cambridge University Press.CrossRefGoogle Scholar
[3] Feldheim, O. N. and Peled, R. Rigidity of 3-colorings of the discrete torus. Ann. Inst. H. Poincaré (B), to appear.Google Scholar
[4] Feldheim, O. N. and Spinka, Y. Long-range order in the 3-state antiferromagnetic Potts model in high dimensions. J. Eur. Math. Soc. (JEMS), to appear.Google Scholar
[5] Galvin, D. (2003) On homomorphisms from the Hamming cube to Z . Israel J. Math. 138 189–213.CrossRefGoogle Scholar
[6] Galvin, D. (2007) Sampling 3-colourings of regular bipartite graphs. Electron. J. Probab. 12 481–497.Google Scholar
[7] Galvin, D. (2008) Sampling independent sets in the discrete torus. Random Struct. Alg. 33 356–376.Google Scholar
[8] Galvin, D. and Kahn, J. (2004) On phase transition in the hard-core model on ℤ d . Combin. Probab. Comput. 13 137–164.Google Scholar
[9] Galvin, D. and Randall, D. (2007) Torpid mixing of local Markov chains on 3-colorings of the discrete torus. In SODA 2007: Proc. 18th Annual ACM–SIAM Symposium on Discrete Algorithms, SIAM, pp. 376–384.Google Scholar
[10] Galvin, D. and Tetali, P. (2004) Slow mixing of Glauber dynamics for the hard-core model on the hypercube. In SODA 2004: Proc. 15th Annual ACM–SIAM Symposium on Discrete Algorithms, SIAM, pp. 466–467.Google Scholar
[11] Galvin, D. and Tetali, P. (2006) Slow mixing of Glauber dynamics for the hard-core model on regular bipartite graphs. Random Struct. Alg. 28 427–443.CrossRefGoogle Scholar
[12] Galvin, D., Kahn, J., Randall, D. and Sorkin, G. (2015) Phase coexistence and torpid mixing in the 3-coloring model on ℤ d . SIAM J. Discrete Math. 29 1223–1244.Google Scholar
[13] Korshunov, A. D. (1981) On the number of monotone Boolean functions. Problemy Kibernet. 38 5–108.Google Scholar
[14] Korshunov, A. D. and Sapozhenko, A. A. (1983) The number of binary codes with distance 2 (in Russian). Problemy Kibernet. 40 111–130.Google Scholar
[15] Lebowitz, J. L. and Mazel, A. E. (1998) Improved Peierls argument for high-dimensional Ising models. J. Statist. Phys. 90 1051–1059.Google Scholar
[16] Lovász, L. (1975) On the ratio of optimal integral and fractional covers. Discrete Math. 13 383–390.Google Scholar
[17] Miranda, Y. M. and Slade, G. (2011) The growth constants of lattice trees and lattice animals in high dimensions. Electron. Comm. Probab. 16 129–136.Google Scholar
[18] Peled, R. (2017) High-dimensional Lipschitz functions are typically flat. Ann. Probab. 45 1351–1447.Google Scholar
[19] Peled, R. and Samotij, W. (2014) Odd cutsets and the hard-core model on ℤ d . Ann. Inst. Henri Poincaré B 50 975–998.CrossRefGoogle Scholar
[20] Sapozhenko, A. A. (1987) On the number of connected subsets with given cardinality of the boundary in bipartite graphs (in Russian). Metody Diskretnogo Analiza. 45 42–70.Google Scholar
[21] Sapozhenko, A. A. (1989) The number of antichains in ranked partially ordered sets. Diskretnaya Matematika 1 74–93.Google Scholar
[22] Sapozhenko, A. A. (1991) On the number of antichains in multilevelled ranked posets. Discrete Math. Appl. 1 149–170.Google Scholar
[23] Timár, Á. (2013) Boundary-connectivity via graph theory. Proc. Amer. Math. Soc. 141 475–480.Google Scholar