02-517 Marek Biskup, Lincoln Chayes and Roman Kotecky

Critical region for droplet formation in the two-dimensional Ising model (245K, Latex) Dec 14, 02

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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~$\beta>\betac$ and overall magnetization conditioned to take the value $\mstar L^2-2\mstar v_L$, where~$\betac^{-1}$ is the critical temperature,~$\mstar=\mstar(\beta)$ 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~$\Deltac$ and a function~$\lambda_\Delta$ such that the following holds: For~$\Delta<\Deltac$, there are no droplets beyond~$\log L$ scale, while for~$\Delta>\Deltac$, there is a single, Wulff-shaped droplet containing a fraction~$\lambda_\Delta\ge\lamc=2/3$ 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.

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