Borodin-P\'ech\'e fluctuations of the free energy in directed random polymer models

TL;DR

This paper studies the fluctuations of free energy in two directed polymer models within the KPZ universality class, demonstrating convergence to Borodin-Péché deformations of the GUE Tracy-Widom distribution.

Contribution

It introduces new boundary conditions for directed polymers and proves their free energy fluctuations converge to Borodin-Péché distributions, extending the understanding of KPZ universality.

Findings

Fluctuations converge to Borodin-Péché distributions

Results apply to both semi-discrete and continuum polymer models

Fluctuation scale is proportional to the cube root of time

Abstract

We consider two directed polymer models in the Kardar-Parisi-Zhang (KPZ) universality class: the O'Connell-Yor semi-discrete directed polymer with boundary sources and the continuum directed random polymer with (m,n)-spiked boundary perturbations. The free energy of the continuum polymer is the Hopf-Cole solution of the KPZ equation with the corresponding (m,n)-spiked initial condition. This new initial condition is constructed using two semi-discrete polymer models with independent bulk randomness and coupled boundary sources. We prove that the limiting fluctuations of the free energies rescaled by the 1/3rd power of time in both polymer models converge to the Borodin-Peche type deformations of the GUE Tracy-Widom distribution.