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MiS Preprint
110/2002
Critical region for droplet formation in the two-dimensional Ising model
Marek Biskup, Lincoln Chayes and Roman Kotecky
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 $\beta>\beta_c$ and overall magnetization conditioned to take the value $m^* L^2-2 m^* v_L$, where $\beta_c^{-1}$ 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 $v_L^{3/2} L^{-2}$ tends to a definite limit. Specifically, we identify a dimensionless parameter $\Delta$, proportional to this limit, a non-trivial critical value $\Delta_c$ and a function $\lambda_\Delta$ 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 $\lambda_\Delta\ge\lambda_c=2/3$ of the magnetization deficit and there are no other droplets beyond the scale of~$\log L$. Moreover, $\lambda_\Delta$ and $\Delta$ are related via a universal equation that apparently is independent of the details of the system.
