Superconcentration for minimal surfaces in first passage percolation and disordered Ising ferromagnets

• 成果类型: Article
• 署名作者: Dembin, Barbara; Garban, Christophe
• 署名单位: Swiss Federal Institutes of Technology Domain; ETH Zurich; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Ecole Centrale de Lyon; Institut National des Sciences Appliquees de Lyon - INSA Lyon; Universite Claude Bernard Lyon 1; Universite Jean Monnet; Institut Universitaire de France
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-023-01252-2
• 发表日期: 2024
• 页码: 675-702
• 关键词: shape fluctuations; large deviations; maximal flows

摘要

We consider the standard first passage percolation model on Z(d) with a distribution G taking two values 0 < a < b. We study the maximal flow through the cylinder [0, n](d-1) x [0, hn] between its top and bottom as well as its associated minimal surface(s). We prove that the variance of the maximal flow is superconcentrated, i.e. in O( n(d-1) /log n), for h >= h(0) (for a large enough constant h(0) = h(0)(a, b)). Equivalently, we obtain that the ground state energy of a disordered Ising ferromagnet in a cylinder [0, n](d-1) x [0, hn] is superconcentrated when opposite boundary conditions are applied at the top and bottom faces and for a large enough constant h >= h(0) (which depends on the law of the coupling constants). Our proof is inspired by the proof of Benjamini-Kalai-Schramm (Ann Probab 31:1970-1978, 2003). Yet, one major difficulty in this setting is to control the influence of the edges since the averaging trick used in Benjamini et al. (Ann Probab 31:1970-1978, 2003) fails for surfaces. Of independent interest, we prove that minimal surfaces (in the present discrete setting) cannot have long thin chimneys.