Shocks and instability in Brownian last-passage percolation

TL;DR

This paper analyzes the structure of shocks, instability points, and their interrelations in Brownian last-passage percolation, revealing how shock trees differ within instability regions and can reconstruct the instability skeleton.

Contribution

It provides a detailed analysis of shocks and instability in Brownian last-passage percolation, linking these features and showing how to reconstruct instability regions from shock trees.

Findings

Shock trees differ within the instability region.

Shock trees align outside the instability region.

The instability region can be reconstructed from shock trees.

Abstract

For stochastic Hamilton-Jacobi (SHJ) equations, instability points are the space-time locations where two eternal solutions with the same asymptotic velocity differ. Another crucial structure in such equations is shocks, which are the space-time locations where the velocity field is discontinuous. In this work, we provide a detailed analysis of the structure and relationships between shocks, instability, and competition interfaces in the Brownian last-passage percolation model, which serves as a prototype of a semi-discrete inviscid stochastic HJ equation in one space dimension. Among our findings, we show that the shock trees of the two unstable eternal solutions differ within the instability region and align outside of it. Furthermore, we demonstrate that one can reconstruct a skeleton of the instability region from these two shock trees.