Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Vincent Beffara, Corentin Faipeur, Tejas Oke

TL;DR

This work addresses the FK random-cluster model with q<1, where FKG monotonicity fails, by developing enhanced stochastic comparisons and a Glauber-dynamics-based approach to bound FK measures between inhomogeneous percolations. It extends the subcritical and supercritical regimes beyond classical limits and provides a framework to prove strong uniqueness of infinite-volume measures in these extended regions, including planar-duality arguments to access the supercritical regime in 2D. A key outcome is a new bound on the critical point p_c(q) on broad graph classes and the demonstration of strong uniqueness under these extended subcritical and planar-duality regimes. The results hold in all dimensions d≥2 and apply to various infinite graphs, offering a versatile toolkit for analyzing phase transitions in q<1 FK-percolation and related Gibbs measures.

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.

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

TL;DR

This work addresses the FK random-cluster model with q<1, where FKG monotonicity fails, by developing enhanced stochastic comparisons and a Glauber-dynamics-based approach to bound FK measures between inhomogeneous percolations. It extends the subcritical and supercritical regimes beyond classical limits and provides a framework to prove strong uniqueness of infinite-volume measures in these extended regions, including planar-duality arguments to access the supercritical regime in 2D. A key outcome is a new bound on the critical point p_c(q) on broad graph classes and the demonstration of strong uniqueness under these extended subcritical and planar-duality regimes. The results hold in all dimensions d≥2 and apply to various infinite graphs, offering a versatile toolkit for analyzing phase transitions in q<1 FK-percolation and related Gibbs measures.

Abstract

We consider the random cluster model with parameter , 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 (resp. ); 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 and beyond hypercubic lattices.

Paper Structure

This paper contains 12 sections, 11 theorems, 50 equations, 5 figures.

Table of Contents

  1. Introduction and main results
  2. Introduction
  3. Definition of the model and comparison inequalities
  4. Specifications for the Random Cluster model
  5. Phase diagram of the FK model
  6. The classical proof of uniqueness in the subcritical regime
  7. Enhanced comparison inequalities for general finite graphs
  8. Extending comparison inequalities to infinite graphs
  9. Proofs of strong uniqueness
  10. Uniqueness in the subcritical phase
  11. Uniqueness in the supercritical phase through planar duality
  12. Open problems

Key Result

Theorem 1.2

Let $G=(V,E)$ be a finite connected graph with boundary $\partial G$, and $\alpha \in \Pi(\partial G)$ be a boundary condition. Assume that $G \cup \alpha$ is not a tree and contains at least two vertices (after wiring). Then, for all $p\in(0,1)$ and $q\neq1$, there exist $(\varepsilon_e)_{e \in E}

Figures (5)

  • Figure 1: Phase diagram of the FK-percolation model on $\mathbf{Z}^2$.
  • Figure 2: Left: Simulation of percolation at $p = 0.45$. Right: Heat bath simulationof FK model at same $p$, with $q = 0.15$.
  • Figure 3: Phase diagram of the FK model on a graph which satisfies the hypothesis of Theorem \ref{['thm:new_bound_pc(q)']}.
  • Figure 4: Left: a "star-device". Right: a "a square-device". The enhanced edge of the device is represented in unbroken line. In dotted line, we represent the other edges of the device, that ensure the presence of a cutset, or of a path connecting both endpoints, made of auxiliary edges only.
  • Figure 5: Left: covering $\mathbb Z^2$ with star-devices, Right: covering $\mathbb Z^2$ with square-devices.

Theorems & Definitions (29)

  • Remark 1.1
  • Theorem 1.2: Enhanced comparison inequalities
  • Remark 1.3
  • Theorem 1.4: Extended subcritical and supercritical phases
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Proposition 1.8
  • Theorem 1.9: Theorem 5.119 in grimmett2006random
  • Theorem 1.10: Strong uniqueness in the subcritical regime
  • ...and 19 more