# Istvan Prause: Integrability of limit shapes

**Time: **
Tue 2020-10-27 15.15

**Lecturer: **
Istvan Prause, University of Eastern Finland

**Location: **
Zoom, meeting ID: 637 2521 0889

### Abstract

Many models in statistical mechanics exhibit limit shape formation: on the macroscopic scale the random system concentrates onto a fixed deterministic limit. These geometric limit shapes arise as minimisers of gradient variational problems. I’ll describe an integrability principle for gradient variational problems: these can be solved in terms of kappa-harmonic functions, that is functions that are harmonic with respect to a Laplacian with a varying conductivity kappa. In case of the dimer model and the 5-vertex model the kappa-Laplacian can be reduced to the usual Laplacian resulting in explicit representation of limit shapes via harmonic functions. The talk is based on joint work with Rick Kenyon.