Two time distribution in Brownian directed percolation

TL;DR

This paper investigates the joint distribution of two last-passage times in Brownian directed percolation, revealing a non-Fredholm determinant expansion in the scaling limit, highlighting slow decorrelation phenomena.

Contribution

It provides the first rigorous computation of the joint distribution for two last-passage times in this model, introducing a novel determinant expansion approach.

Findings

Derived the limiting joint distribution function in a scaling limit.

Identified a non-Fredholm determinant expansion for the distribution.

Connected results to a similar non-rigorous formula by Dotsenko.

Abstract

In the zero temperature Brownian semi-discrete directed polymer we study the joint distribution of two last-passage times at positions ordered in the time-like direction. This is the situation when we have the slow de-correlation phenomenon. We compute the limiting joint distribution function in a scaling limit. This limiting distribution is given by an expansion in determinants which is not a Fredholm expansion. A somewhat similar looking formula was derived non-rigorously in a related model by Dotsenko.