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

2025/12/18 by Vincent Beffara, Beffara, Vincent, Corentin Faipeur +3
Mathematics · Physics and Astronomy · #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #Random Matrices and Applications

paper · doi:10.48550/arxiv.2512.16486

Abstract

We consider the random cluster model with parameter q&lt;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 ≤ q/(1+q) (resp. p ≥ 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 ≥ 2 and beyond hypercubic lattices.

Citations

• Finitary codings for the random-cluster model and other infinite-range\n monotone models
• Sharpness for Inhomogeneous Percolation on Quasi-Transitive Graphs
• Sharp phase transition for the random-cluster and Potts models via\n decision trees
• Discontinuity of the phase transition for the planar random-cluster and Potts models with q>4
• The self-dual point of the two-dimensional random-cluster model is critical for q≥ 1
• Branching Processes, and Random-Cluster Measures on Trees
• The Random-Cluster Model

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• Maximal clusters in non-critical percolation and related models