# Point-to-line polymers and orthogonal Whittaker functions

## TL;DR

This paper analyzes a log-gamma directed polymer model in various geometries, deriving Laplace transform formulas using orthogonal Whittaker functions, and connects these to last passage percolation problems.

## Contribution

It introduces new formulas for the Laplace transforms of partition functions in point-to-line and related geometries using orthogonal Whittaker functions, linking probabilistic models with special functions.

## Key findings

- Derived Laplace transform formulas in different geometries
- Established connections between Whittaker functions and solvable probabilistic models
- Obtained new last passage percolation formulas in the zero-temperature limit

## Abstract

We study a one dimensional directed polymer model in an inverse-gamma random environment, known as the log-gamma polymer, in three different geometries: point-to-line, point-to-half line and when the polymer is restricted to a half space with end point lying free on the corresponding half line.Via the use of A.N.Kirillov's geometric Robinson-Schensted-Knuth correspondence, we compute the Laplace transform of the partition functions in the above geometries in terms of orthogonal Whittaker functions, thus obtaining new connections between the ubiquitous class of Whittaker functions and exactly solvable probabilistic models. In the case of the first two geometries we also provide multiple contour integral formulae for the corresponding Laplace transforms. Passing to the zero-temperature limit, we obtain new formulae for the corresponding last passage percolation problems with exponential weights.