The limit shape phenomenon in statistical mechanics is a version of the central limit theorem in probability theory. It states that a random variable for a large system becomes deterministic in certain scale and remain random, Gaussian distributed, at a smaller scale. Coin tossing is an example. In statistical mechanics a random variable can be a surface, a configuration of paths, a spin configuration. The characterization of the limit shape in general is very complicated. However for a class of systems known as integrable (also subject of study of integrable probability) this can be done quite explicitly.
The talk will be an overview of this direction and of some recent results. As much as time will permit the relation to geometry and representation theory will be explained.