Random polymers via orthogonal Whittaker and symplectic Schur functions

TL;DR

This thesis establishes new exact formulas connecting integrable probabilistic models of directed polymers and last passage percolation to Whittaker and Schur functions, deriving distributions related to the KPZ universality class.

Contribution

It introduces novel formulas for polymer partition functions and last passage percolation models using orthogonal Whittaker and symplectic Schur functions, expanding the analytical toolkit for KPZ models.

Findings

Derived Laplace transform formulas for polymer models.

Obtained new integral formulas for last passage percolation.

Connected models to Tracy-Widom and Airy distributions.

Abstract

This thesis deals with some $(1+1)$-dimensional lattice path models from the KPZ universality class: the directed random polymer with inverse-gamma weights (known as log-gamma polymer) and its zero temperature degeneration, i.e. the last passage percolation model, with geometric or exponential waiting times. We consider three path geometries: point-to-line, point-to-half-line, and point-to-line with paths restricted to stay in a half-plane. Through exact formulas, we establish new connections between integrable probabilistic models and the ubiquitous Whittaker and Schur functions. More in detail, via the use of A. N. Kirillov's geometric Robinson-Schensted-Knuth (RSK) correspondence, we compute the Laplace transform of the polymer partition functions in the above geometries in terms of orthogonal Whittaker functions. In the case of the first two geometries we also provide multiple contour integral formulas. For the corresponding last passage percolation problems, we obtain new formulas in terms of symplectic Schur functions, both directly via RSK on polygonal arrays and via zero temperature limit from the log-gamma polymer formulas. As scaling limits of the point-to-line and point-to-half-line models with exponential waiting times, we derive Sasamoto's Fredholm determinant formula for the GOE Tracy-Widom distribution, as well as the one-point marginal distribution of the ${\rm Airy}_{2\to1}$ process.