Directed Polymers and KPZ universality: a Geometric Approach

Abstract: The late time behaviour of a large class of randomly growing interfaces was postulated by Kardar, Parisi and Zhang in 1986 to be universally governed by the so-called KPZ equation. A renormalization group analysis of that ill-posed SPDE led to prediction of scaling exponents 1/3 and 2/3 for height fluctuation and correlation length respectively. The last two decades have seen an explosion of rigorous results that have verified this prediction and obtained much finer information, for a number of integrable settings, including certain models of directed polymers. I shall briefly recount this success story and describe more recent efforts in studying the polymer geometry by combining the integrable inputs with probabilistic techniques. If time permits, I shall also discuss some cases where this approach has been useful beyond the exactly solvable regime.