Name
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 $n \times n$ box in the Cartesian lattice $\mathbb{Z}^2$. Our main result is a $O(n^2\log n)$ upper bound for the mixing time at all values of the model parameter $p$ except the critical point $p=p_c(q)$, and for all values of the second model parameter $q\ge 1$. 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 $\mathbb{Z}^2$. 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. |
Year | Venue | DocType |
|---|---|---|
2015 | CoRR | Journal |
Volume | Citations | PageRank |
abs/1510.06762 | 0 | 0.34 |
References | Authors | |
2 | 2 | |
Name | Order | Citations | PageRank |
|---|---|---|---|
| Antonio Blanca | 1 | 14 | 9.74 |
| Alistair Sinclair | 2 | 1506 | 308.40 |