Critical region for droplet formation in the two-dimensional Ising model

2002/12/31 by Marek Biskup, Lincoln Chayes, Roman Kotecky
Mathematics · Physics and Astronomy · #math.PR #cond-mat.stat-mech #math-ph #math.MP #physics.chem-ph #msc:60F10 #msc:82B05 #msc:82B24

paper · pdf · doi:10.1007/s00220-003-0946-x

published as Commun. Math. Phys. 242 (2003), no. 1-2, 137--183 · 48 pages, 2 figures, version to appear in Commun. Math. Phys

arxiv created 2003/09/30 · arxiv updated 2009/11/30

Abstract

We study the formation/dissolution of equilibrium droplets in finite systems at parameters corresponding to phase coexistence. Specifically, we consider the 2D Ising model in volumes of size L², inverse temperature β>\betac and overall magnetization conditioned to take the value \mstar L²-2\mstar v_L, where \betac^-1 is the critical temperature, \mstar=\mstar(β) is the spontaneous magnetization and v_L is a sequence of positive numbers. We find that the critical scaling for droplet formation/dissolution is when v_L^3/2 L^-2 tends to a definite limit. Specifically, we identify a dimensionless parameter Δ, proportional to this limit, a non-trivial critical value \Deltac and a function λ_Δ such that the following holds: For Δ<\Deltac, there are no droplets beyond log L scale, while for Δ>\Deltac, there is a single, Wulff-shaped droplet containing a fraction λ_Δ≥\lamc=2/3 of the magnetization deficit and there are no other droplets beyond the scale of log L. Moreover, λ_Δ and Δ are related via a universal equation that apparently is independent of the details of the system.

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