Scaling limits for non-intersecting polymers and Whittaker measures

TL;DR

This paper investigates the asymptotic behavior of partition functions for non-intersecting polymers in random environments, using combinatorial and probabilistic tools to connect microscopic models with macroscopic limits.

Contribution

It provides a new variational framework for understanding the free energy of non-intersecting polymer models via the geometric RSK correspondence and links to random matrix distributions.

Findings

Explicit representation of partition functions via stochastic interfaces

Connection between zero-temperature limit and Marchenko-Pastur distribution

New derivation of surface tension for the bead model

Abstract

We study the partition functions associated with non-intersecting polymers in a random environment. By considering paths in series and in parallel, the partition functions carry natural notions of subadditivity, allowing the effective study of their asymptotics. For a certain choice of random environment, the geometric RSK correspondence provides an explicit representation of the partition functions in terms of a stochastic interface. Formally this leads to a variational description of the macroscopic behaviour of the interface and hence the free energy of the associated non-intersecting polymer model. At zero temperature we relate this variational description to the Marchenko-Pastur distribution, and give a new derivation of the surface tension of the bead model.