[Paper Review] The phase transitions of the planar random-cluster and Potts models with q larger than 1 are sharp

Hugo Duminil‐Copin, Ioan Manolescu|arXiv (Cornell University)|Sep 12, 2014

Stochastic processes and statistical mechanics13 references4 citations

TL;DR

This paper proves the sharpness of phase transitions in planar random-cluster models with cluster weight $ q \geq 1 $, using sharp threshold techniques and lattice symmetries rather than self-duality. The authors establish that for $ p < p_c $, connection probabilities decay exponentially, while for $ p > p_c $, an infinite cluster almost surely exists, with $ p_c $ uniquely defined for various lattices including square, triangular, and hexagonal graphs.

ABSTRACT

We prove that random-cluster models with q larger than 1 on a variety of planar lattices have a sharp phase transition, that is that there exists some parameter p_c below which the model exhibits exponential decay and above which there exists a.s. an infinite cluster. The result may be extended to the Potts model via the Edwards-Sokal coupling. Our method is based on sharp threshold techniques and certain symmetries of the lattice; in particular it makes no use of self-duality. Part of the argument is not restricted to planar models and may be of some interest for the understanding of random-cluster and Potts models in higher dimensions. Due to its nature, this strategy could be useful in studying other planar models satisfying the FKG lattice condition and some additional differential inequalities.

Motivation & Objective

• To establish the sharpness of the phase transition in planar random-cluster models with $ q \geq 1 $, where the system transitions abruptly from exponential decay of connections to the existence of an infinite cluster.
• To provide a proof of sharpness that does not rely on self-duality, unlike prior works, instead using lattice symmetries and sharp threshold techniques.
• To extend the result to weighted biperiodic graphs and to demonstrate that the critical point $ p_c $ coincides with the threshold where exponential decay ceases.
• To show that the critical parameters for standard lattices (square, triangular, hexagonal) are uniquely determined and match known self-dual expressions.
• To lay a foundation for generalizing the method to other planar models satisfying the FKG inequality and differential inequalities.

Proposed method

• The proof employs sharp threshold techniques based on the differential inequalities derived from Russo-type formulas for increasing events.
• It uses the FKG lattice condition to ensure positive association and stochastic ordering of measures under different boundary conditions.
• The argument relies on the domain Markov property to control conditional probabilities and ensure consistency in infinite-volume limits.
• Symmetries of the lattice—specifically reflection and rotational invariance—allow the construction of critical rectangles and dual paths essential for contradiction arguments.
• A key step involves proving that if $ p < p_c $, then dual-cluster connections decay exponentially, which contradicts the existence of a critical point beyond which infinite clusters emerge.
• The method avoids self-duality by using the duality of the random-cluster model and the behavior of dual measures to identify $ p_c $ as the threshold where exponential decay fails.

Experimental results

Research questions

• RQ1 Does the phase transition in planar random-cluster models with $ q \geq 1 $ exhibit sharpness, i.e., is there a unique $ p_c $ such that exponential decay holds below and an infinite cluster exists above?
• RQ2 Can the sharpness of the phase transition be proven without relying on self-duality, which was used in earlier works on the square, triangular, and hexagonal lattices?
• RQ3 What is the critical threshold $ p_c(q) $ for the random-cluster model on the square, triangular, and hexagonal lattices, and can it be explicitly characterized?
• RQ4 To what extent can the method used here be generalized to other planar models satisfying the FKG inequality and differential inequalities?
• RQ5 Is the critical point $ p_c $ equal to the threshold where the probability of connection across large rectangles ceases to decay exponentially?

Key findings

• The phase transition in planar random-cluster models with $ q \geq 1 $ is sharp: for $ p < p_c $, connection probabilities decay exponentially with distance, and for $ p > p_c $, an infinite cluster exists almost surely.
• For the square lattice, $ p_c(q) = \sqrt{q}/(1 + \sqrt{q}) $, which matches the self-dual value, confirming the critical point via the new method.
• For the triangular lattice, $ p_c(q) $ is the unique solution in $[0,1]$ to $ p^3 + 3p^2(1-p) = q(1-p)^3 $, derived without integrability assumptions.
• For the hexagonal lattice, $ p_c(q) $ is the unique solution to $ p^3 - 3qp(1-p)^2 = q^2(1-p)^3 $, again without relying on integrability.
• The method establishes that $ p_c $ is equal to the threshold $ \tilde{p}_c $ where the exponential decay rate of connection probabilities across large rectangles becomes positive.
• The proof shows that if $ p < p_c $, then dual-cluster connections decay exponentially, which contradicts the existence of an infinite cluster at $ p > p_c $, confirming sharpness.