Random-cluster dynamics in ℤ2

Antonio Blanca Pimentel, Alistair Sinclair

Research output: Chapter in Book/Report/Conference proceedingConference contribution

2 Citations (Scopus)

Abstract

The random-cluster model has been widely studied as a unifying framework for random graphs, spin systems and electrical networks, but its dynamics have so far largely resisted analysis. In this paper we analyze the Glauber dynamics of the random-cluster model in the canonical case where the underlying graph is an n × n box in the Cartesian lattice ℤ2. Our main result is a O(n2 log n) upper bound for the mixing time at all values of the model parameter p except the critical point p = pc(q), and for all values of the second model parameter q ≤ 1. We also provide a matching lower bound proving that our result is tight. Our analysis takes as its starting point the recent breakthrough by Beffara and Duminil-Copin on the location of the random-cluster phase transition in ℤ2. It is reminiscent of similar results for spin systems such as the Ising and Potts models, but requires the reworking of several standard tools in the context of the random-cluster model, which is not a spin system in the usual sense.

Original languageEnglish (US)
Title of host publication27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016
EditorsRobert Krauthgamer
PublisherAssociation for Computing Machinery
Pages498-513
Number of pages16
ISBN (Electronic)9781510819672
StatePublished - Jan 1 2016
Event27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016 - Arlington, United States
Duration: Jan 10 2016Jan 12 2016

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms
Volume1

Conference

Conference27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016
CountryUnited States
CityArlington
Period1/10/161/12/16

Fingerprint

Random-cluster Model
Spin Systems
Electrical Networks
Glauber Dynamics
Mixing Time
Potts Model
Cartesian
Random Graphs
Ising Model
Potts model
Critical point
Phase Transition
Ising model
Lower bound
Upper bound
Graph in graph theory
Model
Phase transitions

All Science Journal Classification (ASJC) codes

  • Software
  • Mathematics(all)

Cite this

Pimentel, A. B., & Sinclair, A. (2016). Random-cluster dynamics in ℤ2. In R. Krauthgamer (Ed.), 27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016 (pp. 498-513). (Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms; Vol. 1). Association for Computing Machinery.
Pimentel, Antonio Blanca ; Sinclair, Alistair. / Random-cluster dynamics in ℤ2. 27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016. editor / Robert Krauthgamer. Association for Computing Machinery, 2016. pp. 498-513 (Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms).
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Pimentel, AB & Sinclair, A 2016, Random-cluster dynamics in ℤ2. in R Krauthgamer (ed.), 27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016. Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms, vol. 1, Association for Computing Machinery, pp. 498-513, 27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, United States, 1/10/16.

Random-cluster dynamics in ℤ2. / Pimentel, Antonio Blanca; Sinclair, Alistair.

27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016. ed. / Robert Krauthgamer. Association for Computing Machinery, 2016. p. 498-513 (Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms; Vol. 1).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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Pimentel AB, Sinclair A. Random-cluster dynamics in ℤ2. In Krauthgamer R, editor, 27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016. Association for Computing Machinery. 2016. p. 498-513. (Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms).