A new bound for the critical point of the FK model for $q<1$

TL;DR

This paper extends bounds on the critical point of the FK model for q<1, using modified Glauber dynamics to improve stochastic comparisons and establish measure uniqueness across various dimensions.

Contribution

It introduces a novel approach with modified Glauber dynamics to refine bounds on the critical point of the FK model for q<1, expanding the known parameter ranges.

Findings

Improved stochastic bounds for the FK model with q<1.

Proof of uniqueness of the infinite-volume measure in extended ranges.

Results applicable in any dimension d ≥ 2 and beyond hypercubic lattices.

Abstract

We consider the random cluster model with parameter $q<1$, for which the FKG inequalities are not valid. On the square lattice, stochastic comparison with Bernoulli percolation implies that the model is subcritical (respectively supercritical) when $p \leq q/(1+q)$ (resp. $p \geq 1/2$); in this paper, we extend these two regions, by improving the classical stochastic comparisons. Assuming the existence of the critical point, this reduces its possible range. The proof relies on a modification of the usual Glauber dynamics of the model, which enables stochastic bounds of FK measures between two inhomegenous percolations. We also prove uniqueness of the infinite-volume measure in our extended ranges. Most of our results are valid in any dimension $d \geq 2$ and beyond hypercubic lattices.