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

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.