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

TL;DR

This paper establishes the precise point of phase transition in the random-cluster model with summable external magnetic fields on hypercubic lattices, showing a clear change from decay to percolation.

Contribution

It proves the sharpness of phase transition for the random-cluster model with external fields, extending understanding to models with cluster weight q and summable positive external fields.

Findings

Existence of a nontrivial critical parameter for phase transition.

Below the critical point, the model exhibits exponential decay.

Above the critical point, an infinite cluster almost surely exists.

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=2,3,..., on the hypercubic lattice. That is, there exists some nontrivial critical parameter that depends 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.