Symmetric polynomials in Integrable Probability

Abstract: "A number of solvable stochastic processes can be described in terms of notable families of symmetric functions. Classical models as the last passage percolation (LPP) or the totally asymmetric simple exclusion process (TASEP) sample measures built on Schur polynomials. Analogously, Whittaker functions are related to solvable models of random polymers as the O’Connell-Yor Polymer (OYP).

In 2015 Corwin and Petrov introduced the higher spin vertex model, a family of stochastic processes sitting on top of a hierarchy of models including TASEP, LPP, OYP and of many other interesting systems including random walkers in random environment.

We find that the higher spin vertex model and all of its degenerations can be solved using a unifying family of symmetric functions, the spin q-Whittaker (sqW) polynomials, a version of which was defined first by Borodin and Wheeler in 2017. Probabilistic intepretation of sqW allows us to establish a number of interesting combinatorial properties along with surprising conjectural relations. Studying scaling limits of sqW we recover classical objects as Schur and Grothendieck polynomials along with new families of symmetric functions.

Based on a joint work with Leonid Petrov."

combinatorics

Audience: researchers in the topic

York University Applied Algebra Seminar