2020/04/22 by Shirshendu Ganguly, Reza Gheissari, Ganguly, Shirshendu +1
Mathematics · Physics and Astronomy · #FOS: Mathematics #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Mathematical Physics (math-ph) #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.2004.10737
openalex publication_date 2020/04/22 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
Consider the Ising model at low-temperatures and positive external field\n\λ on an N\× N box with Dobrushin boundary conditions that are\nplus on the north, east, and west boundaries and minus on the south boundary.\nIf \λ = 0, the interface separating the plus and minus phases is\ndiffusive, having O(\√ N) height fluctuations, and the model is fully\nwetted. Under an order one field, the interface fluctuations are O(1) and the\ninterface is only partially wetted, being pinned to its southern boundary. We\nstudy the critical pre-wetting regime of \λN downarrow 0, where the\nheight fluctuations are expected to scale as \λ -1/3 and the rescaled\ninterface is predicted to converge to the Ferrari--Spohn diffusion. Velenik\n(2004) identified the order of the area under the interface up to logarithmic\ncorrections. Since then, more refined features of such interfaces have only\nbeen identified in simpler models of random walks under area tilts.\n In this paper, we resolve several conjectures of Velenik regarding the\nrefined features of the Ising interface in the critical pre-wetting regime. Our\nmain result is a sharp bound on the one-point height fluctuation, proving e^\n- \Θ(x3/2) upper tails reminiscent of the Tracy--Widom distribution,\ncapturing a tradeoff between the locally Brownian oscillations and the global\nfield effect. We further prove a concentration estimate for the number of\npoints above which the interface attains a large height. These are used to\ndeduce various geometric properties of the interface, including the order and\ntails of the area it confines, and the poly-logarithmic pre-factor governing\nits maximum height fluctuation. Our arguments combine classical inputs from the\nrandom-line representation of the Ising interface, with novel local resampling\nand coupling schemes.\n