Marek Biskup, Lincoln Chayes and Roman Kotecky
Critical region for droplet formation in the two-dimensional Ising model
(245K, Latex)

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.