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

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 beta > beta(c) and overall magnetization conditioned to take the value m*L(2) - 2m*v(L), where beta(c)(-l) is the critical temperature, m* = m*(beta) 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(K)(3/2)L(-2) tends L to a definite limit. Specifically, we identify a dimensionless parameter A, proportional to this limit, a non-trivial critical value Delta(c) and a function lambda(A) such that the following holds: For Delta < Delta(c), there are no droplets beyond log L scale, while for Delta > Delta(c), there is a single, Wulff-shaped droplet containing a fraction lambda(Delta) greater than or equal to lambda(c) = 2/3 of the magnetization deficit and there are no other droplets beyond the scale of log L. Moreover, lambda(A) and Delta are related via a universal equation that apparently is independent of the details of the system.