Schur generating functions for 2d probability and asymptotic representation theory

Many systems of 2d statistical mechanics and asymptotic representation theory lead to random ordered N-tuples of integers. I will explain an approach for studying the
asymptotic macroscopic properties of such N-tuples as N tends to infinity. A central tool is the Schur generating function, which replaces characteristic functions (or Fourier transforms) in this context. The approach leads to exact asymptotic results for random lozenge and domino tilings, families of non-intersecting paths, as well as tensor products and restrictions of the irreducible representations of classical Lie groups.