A Short Proof of the Discontinuity of Phase Transition in the Planar Random-Cluster Model with $$q>4$$ q > 4
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Communications in
Mathematical Physics
A Short Proof of the Discontinuity of Phase Transition in the Planar Random-Cluster Model with q > 4 Gourab Ray1 , Yinon Spinka2 1 Department of Mathematics, University of Victoria, Victoria, BC V8W 2Y2, Canada.
E-mail: [email protected]
2 Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada.
E-mail: [email protected] Received: 1 June 2019 / Accepted: 14 December 2019 Published online: 24 August 2020 – © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract: The goal of this paper is to provide a short proof of the discontinuity of phase transition for the random-cluster model on the square lattice with parameter q > 4. This result was recently shown in Duminil-Copin et al. (arXiv:1611.09877, 2016) via the so-called Bethe ansatz for the six-vertex model. Our proof also exploits the connection to the six-vertex model, but does not rely on the Bethe ansatz. Our argument is soft (in particular, it does not rely on a computation of the correlation length) and only uses very basic properties of the random-cluster model [for example, we do not even need the Russo–Seymour–Welsh machinery developed recently in Duminil-Copin et al. (Commun Math Phys 349(1):47–107, 2017)]. 1. Introduction The random-cluster model is a well-known and, by now, also a well-studied dependent percolation model. Suppose we are given a finite graph G and two parameters p ∈ [0, 1] and q > 0, called the edge weight and cluster weight. The random-cluster model is a probability measure on {0, 1} E(G) which assigns to a configuration ω a probability proportional to p o(ω) (1 − p)c(ω) q k(ω) , where o(ω) is the number of open edges (edges with value 1), c(ω) = |E(G)| − o(ω) is the number of closed edges, and k(ω) is the number of vertex clusters in ω. In this paper, we are concerned with the random-cluster model on the square lattice with parameter q ≥ 1. As this is an infinite graph, one must take an appropriate limit in the definition above. The random-cluster model satisfies a monotonicity (FKG) property that gives rise two natural limits called the free random-cluster measure and the wired random-cluster measure. We denote these measures by Pfp,q and Pwp,q , respectively. Gourab Ray: Supported in part by NSERC 50311-57400 and University of Victoria start-up 10000-27458.
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G. Ray, Y. Spinka
Both measures are translation-invariant probability measures on {0, 1} E(Z ) which are extremal in a certain sense. An important quantity in this model is the probability that the origin belongs to an infinite cluster. Let θ f ( p, q) and θ w ( p, q) denote these probabilities under Pfp,q and Pwp,q , respectively. It is well-known that the random-cluster model undergoes a phase transition as p varies in the sense that, for any q ≥ 1 there exists a critical threshold pc = pc (q) ∈ (0, 1) such that θ f ( p, q) and θ w ( p, q) are both 0 for all p < pc and are both positive for all p > pc . In fact, √in the case √ of the square lattice, it has been shown [4] that the cr
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