Upper bounds on the percolation correlation length

2019/02/08 by Hugo Duminil-Copin, Hugo Duminil‐Copin, Duminil-Copin, Hugo +4 · 3 citations
Mathematics · #60K35 (primary) #68Q87 (secondary) #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics #math.PR #msc:60K35 #msc:68Q87

paper · pdf · doi:10.48550/arxiv.1902.03207

21 pages, 2 figures

openalex publication_date 2019/02/08 · arxiv created 2020/02/06 · arxiv updated 2020/02/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the size of the near-critical window for Bernoulli percolation on \mathbb Z^d. More precisely, we use a quantitative Grimmett-Marstrand theorem to prove that the correlation length, both below and above criticality, is bounded from above by exp(C/|p-p_c|²). Improving on this bound would be a further step towards the conjecture that there is no infinite cluster at criticality on \mathbb Z^d for every d≥2.

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