# Point-to-line last passage percolation and the invariant measure of a system of reflecting Brownian motions

## TL;DR

This paper establishes a deep connection between the invariant measure of reflected Brownian motions and point-to-line last passage percolation times, revealing new insights into stochastic processes and directed polymers.

## Contribution

It proves an equality in law linking invariant measures of reflected Brownian motions with point-to-line last passage percolation times, and introduces a novel interacting Brownian system with a log partition function invariant measure.

## Key findings

- Invariant measure of reflected Brownian motions equals point-to-line last passage percolation times.
- Distribution of Dyson Brownian motion's supremum with drift is characterized.
- Finite temperature version relates two directed polymer models and generalizes Dufresne's identity.

## Abstract

This paper proves an equality in law between the invariant measure of a reflected system of Brownian motions and a vector of point-to-line last passage percolation times in a discrete random environment. A consequence describes the distribution of the all-time supremum of Dyson Brownian motion with drift. A finite temperature version relates the point-to-line partition functions of two directed polymers, with an inverse-gamma and a Brownian environment, and generalises Dufresne's identity. Our proof introduces an interacting system of Brownian motions with an invariant measure given by a field of point-to-line log partition functions for the log-gamma polymer.

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Source: https://tomesphere.com/paper/1904.03253