# Mathematics Colloquia and Seminars

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### Critical scale for droplet formation in the 2D Ising model

**Mathematical Physics & Probability**

Speaker: | Dr. Marek Biskup, UCLA |

Location: | 493 Kerr |

Start time: | Tue, Nov 12 2002, 3:10PM |

I will discuss the ctitical scale for appearance/disappearance of a droplet in the Ising model with a magnetization fixed to a number slightly below the equilibrium value. (This corresponds to the canonical distribution for the associated lattice-gas model with a slight exess of particles) Based on simple large-deviation analysis (whose validity seems to extend far beyond the model and the dimension at hand), I will dervive a universal scaling relation for the size of the droplet and show the existence of a minimal droplet size beyond which any droplet prefers to melt and dissolve into the background. This result clarifies the cases that were left open in previous rigorous treatments of the Wulff construction. The developed techniques also offer the possibility to give a rigorous proof of the classic Gibbs-Thomson formula for the excess of the pressure/density due to the presence of a curved interface.