MiS Preprint Repository

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 $m^{\ast }{L}^2-2m^{\ast }{v}_L$, where ${\beta }_c^{-1}$ is the critical temperature, $m^{\ast }=m^{\ast }(\beta )$ 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 $\mathrm{\Delta }$, proportional to this limit, a non-trivial critical value ${\mathrm{\Delta }}_c$ and a function ${\lambda }_{\mathrm{\Delta }}$ such that the following holds: For $\mathrm{\Delta }<{\mathrm{\Delta }}_c$, there are no droplets beyond $\log L$ scale, while for $\mathrm{\Delta }>{\mathrm{\Delta }}_c$, there is a single, Wulff-shaped droplet containing a fraction ${\lambda }_{\mathrm{\Delta }}\ge {\lambda }_c=2/3$ of the magnetization deficit and there are no other droplets beyond the scale of~$\log L$. Moreover, ${\lambda }_{\mathrm{\Delta }}$ and $\mathrm{\Delta }$ are related via a universal equation that apparently is independent of the details of the system.