Cookies

We use cookies to ensure that we give you the best experience on our website. By continuing to browse this repository, you give consent for essential cookies to be used. You can read more about our Privacy and Cookie Policy.


Durham Research Online
You are in:

Finite-size scaling, phase coexistence, and algorithms for the random cluster model on random graphs

Helmuth, Tyler and Jenssen, Matthew and Perkins, Will (2022) 'Finite-size scaling, phase coexistence, and algorithms for the random cluster model on random graphs.', Annales de l'Institut Henri Poincaré, Probabilités et Statistiques .

Abstract

For ∆ ≥ 5 and q large as a function of ∆, we give a detailed picture of the phase transition of the random cluster model on random ∆-regular graphs. In particular, we determine the limiting distribution of the weights of the ordered and disordered phases at criticality and prove exponential decay of correlations and central limit theorems away from criticality. Our techniques are based on using polymer models and the cluster expansion to control deviations from the ordered and disordered ground states. These techniques also yield efficient approximate counting and sampling algorithms for the Potts and random cluster models on random ∆-regular graphs at all temperatures when q is large. This includes the critical temperature at which it is known the Glauber and Swendsen-Wang dynamics for the Potts model mix slowly. We further prove new slowmixing results for Markov chains, most notably that the Swendsen-Wang dynamics mix exponentially slowly throughout an open interval containing the critical temperature. This was previously only known at the critical temperature. Many of our results apply more generally to ∆-regular graphs satisfying a small-set expansion condition.

Item Type:Article
Full text:(AM) Accepted Manuscript
Download PDF
(477Kb)
Status:Peer-reviewed
Publisher Web site:https://projecteuclid.org/journals/annales-de-linstitut-henri-poincare-probabilites-et-statistiques
Date accepted:17 March 2022
Date deposited:12 October 2022
Date of first online publication:No date available
Date first made open access:12 October 2022

Save or Share this output

Export:
Export
Look up in GoogleScholar