Directed polymers and the quantum Toda lattice

TL;DR

This paper links the partition function of a Brownian directed polymer to a diffusion process related to the quantum Toda lattice, using a geometric RSK correspondence and extending Matsumoto and Yor's theorem.

Contribution

It introduces a novel characterization of the polymer partition function through a diffusion process connected to the quantum Toda lattice, generalizing existing theorems.

Findings

Partition function described by a diffusion process linked to quantum Toda lattice

Extended Matsumoto and Yor's theorem to multidimensional exponential functionals

Established a geometric RSK correspondence variant for this context

Abstract

We characterize the law of the partition function of a Brownian directed polymer model in terms of a diffusion process associated with the quantum Toda lattice. The proof is via a multidimensional generalization of a theorem of Matsumoto and Yor concerning exponential functionals of Brownian motion. It is based on a mapping which can be regarded as a geometric variant of the RSK correspondence.