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 variational problems which results in explicit representation of limit shapes in terms of harmonic functions across a variety of models. Some new features beyond free fermionic settings will be discussed. 

The talk is based on joint works with Rick Kenyon (https://arxiv.org/abs/2006.01219 and https://arxiv.org/abs/2101.04195).