Bounds for scaling exponents for a 1+1 dimensional directed polymer in a Brownian environment

TL;DR

This paper determines the exact scaling exponents for a 1+1-dimensional directed polymer in a Brownian environment with stationary boundary conditions and provides conjectured bounds for the non-boundary case.

Contribution

It precisely identifies the exponents for the stationary boundary case and proposes conjectured bounds for the non-boundary case of the directed polymer model.

Findings

Exact exponents identified for stationary boundary conditions.

Conjectured upper bounds established for non-boundary conditions.

Advances understanding of scaling behavior in directed polymers.

Abstract

We study the scaling exponents of a 1+1-dimensional directed polymer in a Brownian random environment introduced by O'Connell and Yor. For a version of the model with boundary conditions that are stationary in a space-time sense we identify the exact values of the exponents. For the version without the boundary conditions we get the conjectured upper bounds on the exponents.