A finite-volume version of Aizenman-Higuchi theorem for the 2d Ising model

摘要

In the late 1970s, in two celebrated papers, Aizenman and Higuchi independently established that all infinite-volume Gibbs measures of the two-dimensional ferromagnetic nearest-neighbor Ising model at inverse temperature β ≥ 0 are of the form α μ ~+ _β + (1 - α) μ ~- _β, where μ ~+ _β and μ - β and μ ~- _β are the two pure phases and 0 ≤ α ≤ 1. We present here a new approach to this result, with a number of advantages: (a) We obtain an optimal finite-volume, quantitative analogue (implying the classical claim); (b) the scheme of our proof seems more natural and provides a better picture of the underlying phenomenon; (c) this new approach might be applicable to systems for which the classical method fails.