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

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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> and overall magnetization conditioned to take the value L²-2 vₗ, where ⁻¹ is the critical temperature, =(β) is the spontaneous magnetization and vₗ is a sequence of positive numbers. We find that the critical scaling for droplet formation/dissolution is when vₗ³/² L⁻² tends to a definite limit. Specifically, we identify a dimensionless parameter Δ, proportional to this limit, a non-trivial critical value and a function λΔ such that the following holds: For Δ<, there are no droplets beyond L scale, while for Δ>, there is a single, Wulff-shaped droplet containing a fraction λΔ≥=2/3 of the magnetization deficit and there are no other droplets beyond the scale of L. Moreover, λΔ and Δ are related via a universal equation that apparently is independent of the details of the system.