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Critical Region for Droplet Formation in the Two-Dimensional Ising Model

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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 L 2, inverse temperature β>β c and overall magnetization conditioned to take the value m L 2−2m v L , where β c −1 is the critical temperature, m =m (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 Δ c and a function λΔ such that the following holds: For Δ<Δ c , there are no droplets beyond log L scale, while for Δ>Δ c , there is a single, Wulff-shaped droplet containing a fraction λΔ≥λ c =2/3 of the magnetization deficit and there are no other droplets beyond the scale of log L. Moreover, λΔ and Δ are related via a universal equation that apparently is independent of the details of the system.

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Biskup, M., Chayes, L. & Kotecký, R. Critical Region for Droplet Formation in the Two-Dimensional Ising Model. Commun. Math. Phys. 242, 137–183 (2003). https://doi.org/10.1007/s00220-003-0946-x

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