Planar random-cluster model: scaling relations

2020/11/30 by Hugo Duminil‐Copin, Duminil-Copin, Hugo, Ioan Manolescu +1 · 3 citations
Mathematics · #FOS: Mathematics #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Mathematical Physics (math-ph) #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2011.15090

openalex publication_date 2020/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper studies the critical and near-critical regimes of the planar random-cluster model on \mathbb Z² with cluster-weight q∈[1,4] using novel coupling techniques. More precisely, we derive the scaling relations between the critical exponents β, γ, δ, η, ν, ζ as well as α (when α≥0). As a key input, we show the stability of crossing probabilities in the near-critical regime using new interpretations of the notion of influence of an edge in terms of the rate of mixing. As a byproduct, we derive a generalization of Kesten's classical scaling relation for Bernoulli percolation involving the ``mixing rate'' critical exponent ι replacing the four-arm event exponent ξ₄.

Cited by

• A simple proof of exponential decay for the near-critical planar Ising model
• Liouville quantum gravity: from random planar maps to conformal field theory
• Arm events in critical planar loop soups

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