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

Helmuth, Tyler; Jenssen, Matthew; Perkins, Will

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## Authors

**Dr Tyler Helmuth** tyler.helmuth@durham.ac.uk

Associate Professor
Matthew Jenssen

Will Perkins

**Abstract**

For $\Delta \ge 5$ and $q$ large as a function of $\Delta$, we give a detailed picture of the phase transition of the random cluster model on random $\Delta$-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 $\Delta$-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 $\Delta$-regular graphs satisfying a small-set expansion condition.

**Citation**

Helmuth, T., Jenssen, M., & Perkins, W. (2023). Finite-size scaling, phase coexistence, and algorithms for the random cluster model on random graphs. Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 59(2), 817-848. https://doi.org/10.1214/22-aihp1263

**Journal** Article
**Article Type**

**Acceptance** Mar 17, 2022
**Date**
**Online** Apr 12, 2023

**Publication Date**

**Publication Date** 2023-05

**Deposit Date** Mar 18, 2022

**Publicly Available Date** Oct 12, 2022

**Journal** Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

**Print ISSN** 0246-0203

**Publisher** Institute of Mathematical Statistics

**Peer Reviewed**

**Volume** 59

**Issue** 2

**Pages** 817-848

**DOI** https://doi.org/10.1214/22-aihp1263

**Public URL** https://durham-repository.worktribe.com/output/1211220

**Related Public URLs** https://arxiv.org/abs/2006.11580

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**Accepted Journal Article** (488 Kb)

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