Determinantal structures in the O'Connell-Yor directed random polymer model

TL;DR

This paper explores the semi-discrete directed random polymer model by deriving a determinantal measure representation for its partition function, linking it to random matrix theory and extending GUE eigenvalue measures.

Contribution

It introduces a novel determinantal measure for the polymer model's partition function, connecting it to GUE eigenvalue distributions and expanding the understanding of its probabilistic structure.

Findings

Representation of the moment generating function as a determinantal measure

Extension of GUE eigenvalue measure to the polymer model

Introduction of a larger determinantal measure with key properties

Abstract

We study the semi-discrete directed random polymer model introduced by O'Connell and Yor. We obtain a representation for the moment generating function of the polymer partition function in terms of a determinantal measure. This measure is an extension of the probability measure of the eigenvalues for the Gaussian Unitary Ensemble (GUE) in random matrix theory. To establish the relation, we introduce another determinantal measure on larger degrees of freedom and consider its few properties, from which the representation above follows immediately.