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MiS Preprint

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

Marek Biskup, Lincoln Chayes and Roman Kotecky


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.

droplet formation, ising model, wulff shape

Related publications

2003 Repository Open Access
Roman Kotecký, Marek Biskup and Lincoln Chayes

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

In: Communications in mathematical physics, 242 (2003) 1/2, pp. 137-183