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Istvan Prause: Integrability of limit shapes

Time: Tue 2020-10-27 15.15

Location: Zoom, meeting ID: 637 2521 0889

Participating: Istvan Prause, University of Eastern Finland


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.