2025/12/11 by Pierre‐François Rodriguez, Rodriguez, Pierre-François, Zhang, Wen
Mathematics · Physics and Astronomy · #60G15 #60J45 #60K35 #82B43 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.2512.10933
openalex publication_date 2025/12/11 · openalex created_date 2025/12/13 · openalex updated_date 2026/07/28
We consider the Gaussian free field φ on ℤ2 at large spatial scales N and give sharp bounds on the probability θ(a,N) that the radius of a finite cluster in the excursion set \φ≥ a\ on the corresponding metric graph is macroscopic. We prove a scaling law for this probability, by which θ(a,N) transitions from fractional logarithmic decay for near-critical parameters (a,N) to polynomial decay in the off-critical regime. The transition occurs across a certain scaling window determined by a correlation length scale ξ, which is such that θ(a,N) ∼ θ(0,ξ)(\tfracNξ)-τ for typical heights a as N/ξ diverges, with an explicit exponent τ that we identify in the process. This is in stark contrast with recent results from arXiv:2101.02200 and arXiv:2312.10030 in dimension three, where similar observables are shown to follow regular scaling laws, with polynomial decay at and near criticality, and rapid decay in N/ξ away from it.