2024/12/23 by Tobias Ried, Ried, Tobias, Christian Wagner +1
Mathematics · #49Q05 #60K35 #82B44 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Point processes and geometric inequalities #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2412.17625
openalex publication_date 2024/12/23 · openalex created_date 2024/12/25 · openalex updated_date 2026/07/28
We consider a model of random curves in the plane related to the large-scale behavior of the Random Field Ising Model (RFIM) at temperature zero in two space dimensions. Our work is motivated by attempts to quantify the Imry-Ma phenomenon concerning the rounding of the phase transition by quenched disorder, and connects to recent advances regarding the decay of correlations in the RFIM. We study a continuum model of minimal surfaces in two space dimensions subject to an external, quenched random field, and restrict ourselves to isotropic surface integrands. The random fields we consider behave like white noise on large scales with an ultra-violet regularization reminiscent of the lattice structure of the RFIM. We give a finer description of the minimizer below the length scale L_* starting from which the influence of boundary conditions is suppressed with a given probability, which has recently been shown to satisfy log L_* ∼ ε-(4)/(3) in the amplitude ε>0 of the noise. More precisely, we prove flatness of the phase boundaries on scales L up to log L \lesssim ε-(4)/(13) .