Konstantin Matetski : Polynuclear growth and the Toda lattice

Posted by Cheng Ouyang , part of the Analysis and Applied Mathematics Seminar.

At

Dec. 2, 2024, 4 p.m.

In

636 SEO

Abstract

Polynuclear growth is one of the basic models in the Kardar-Parisi-Zhang universality class, which describes a one-dimensional crystal growth. For a particular initial state, its one-point value equals the length of the longest increasing subsequence for uniformly random permutations (whose asymptotic behavior was first studied by S. Ulam). In my joint work with J. Quastel and D. Remenik, we computed the distribution function of the polynuclear growth with arbitrary initial conditions. These formulas allowed us to express the distribution function in terms of the solutions of the Toda lattice, one of the classical integrable systems. A suitable rescaling of the model yields a non-trivial continuous limit of the polynuclear growth (the KPZ fixed point) and the respective equations (Kadomtsev-Petviashvili).