Search

MiS Preprint Repository

We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
110/2002

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

Marek Biskup, Lincoln Chayes and Roman Kotecky

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 $\beta>\beta_c$ 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.

Received:
Dec 13, 2002
Published:
Dec 13, 2002
Keywords:
droplet formation, ising model, wulff shape

Related publications

inJournal
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