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

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 $β>\betac$ and overall magnetization conditioned to take the value $\mstar L^2-2\mstar v_L$, where $\betac^{-1}$ is the critical temperature, $\mstar=\mstar(β)$ 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 $Δ$, proportional to this limit, a non-trivial critical value $\Deltac$ and a function $λ_Δ$ such that the following holds: For $Δ<\Deltac$, there are no droplets beyond $\log L$ scale, while for $Δ>\Deltac$, there is a single, Wulff-shaped droplet containing a fraction $λ_Δ\ge\lamc=2/3$ of the magnetization deficit and there are no other droplets beyond the scale of $\log L$. Moreover, $λ_Δ$ and $Δ$ are related via a universal equation that apparently is independent of the details of the system.

48 pages, 2 figures, version to appear in Commun. Math. Phys