On the speed of convergence for two-dimensional first passage Ising percolation

• 成果类型: Article
• 署名作者: Higuchi, Y; Zhang, Y
• 署名单位: Kobe University; University of Colorado System; University of Colorado at Colorado Springs
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 2000
• 页码: 353-378

摘要

Consider first passage Ising percolation on Z(2). Let beta denote the reciprocal temperature and let h denote an external magnetic field. Denote by beta(c) the critical temperature and, for beta < beta(c), let h(c)(beta)= h(c) = sup{h: theta(beta, h) = 0}, where theta(beta, h) is the probability that the origin is connected by an infinite (+)-cluster. With these definitions let us consider first passage Ising percolation on Z(2). Let a(0,n) denote the first passage time from (0, 0) to (n, 9). It follows from a subadditive argument that lim(n-->infinity) a(0,n)/n = v a.s and in L-1. It is known that v > 0 if beta < beta(c) and \h\ < h(c)(beta). Here we will estimate the speed of the convergence, vn less than or equal to Ea(0,n) less than or equal to vn + C(n log(5) n)(1/2) for some constant C. Define mu(beta, h) to be the unique Gibbs measure for beta < beta(c). We also prove that there exist (C) over tilde, <(alpha)over tilde> > 0 such that mu(beta, h)(\a(0, n) - Ea(0, n)\ greater than or equal to x) less than or equal to (C) over tilde exp(-<(alpha)over tilde> x(2)/n log(4)n). In addition to a(0, n), we shall also discuss other passage times.