Sharp Phase Transition for the Random-Cluster Model with Summable External Magnetic Field

#### R. Vila

2021, v.27, Issue 1, 43-62

ABSTRACT

In this paper, we prove sharpness of the phase transition for the random-
cluster model in summable positive external fields, with cluster weight $q\in\{2,3,\ldots\}$,
on the hypercubic lattice
$\mathbb{Z}^d$, $d\geqslant 2$.
That is, there exists some critical parameter $0<\beta_c<\infty$ that dep
ends on the cluster weight and the external field, below which the model exhibits exponential decay and above which there exists almost surely an infinite cluster.

Keywords: Random-cluster model, non-translation-invariant external field, sharp phase transition, exponential decay, FKG property