From stability to chaos in last-passage percolation

TL;DR

This paper investigates the transition from stable to chaotic behavior in a dynamic last-passage percolation model on multidimensional integer lattices, revealing a phase transition related to the variance of passage times.

Contribution

It establishes a phase transition criterion between stability and chaos in dynamic last-passage percolation, based on the resampling probability and passage time variance.

Findings

High correlation of passage times at small perturbations

Geodesics become disjoint at large perturbations

Phase transition occurs at t proportional to variance of T

Abstract

We study the transition from stability to chaos in a dynamic last passage percolation model on $\mathbb{Z}^d$ with random weights at the vertices. Given an initial weight configuration at time $0$, we perturb the model over time in such a way that the weight configuration at time $t$ is obtained by resampling each weight independently with probability $t$. On the cube $[0,n]^d$, we study geodesics, that is, weight-maximizing up-right paths from $(0,0, \dots, 0)$ to $(n,n, \dots, n)$, and their passage time $T$. Under mild conditions on the weight distribution, we prove a phase transition between stability and chaos at $t \asymp \frac{1}{n}\mathrm{Var}(T)$. Indeed, as $n$ grows large, for small values of $t$, the passage times at time $0$ and time $t$ are highly correlated, while for large values of $t$, the geodesics become almost disjoint.