Geodesic switches and exceptional times in dynamical Brownian last passage percolation

TL;DR

This paper studies the dynamic behavior of geodesics in Brownian last passage percolation, quantifying the number of switches over time and analyzing the fractal dimensions of exceptional times when special geodesics exist.

Contribution

It provides bounds on the expected number of geodesic switches and characterizes the Hausdorff dimension of times with special geodesic properties in a dynamic percolation model.

Findings

Expected total number of switches is at most n^{5/3+o(1)}(t-s).

Hausdorff dimension of exceptional times with bi-infinite geodesics is at most 1/2.

Dimension of times with directed bi-infinite geodesics is zero.

Abstract

We consider Brownian last passage percolation evolving dynamically via a discrete resampling procedure. Using $\Gamma_{(0,0)}^{(n,n),r}$ to denote a geodesic from $(0,0)$ to $(n,n)$ at time $r$, we prove that the expected total number of coarse-grained changes (or "switches") accumulated by $\Gamma_{(0,0)}^{(n,n),r}$ away from its endpoints during a time interval $[s,t]$ is at most $n^{5/3+o(1)}(t-s)$; we expect the exponent $5/3$ to be tight. Using the above estimate, we establish that the set $\mathscr{T}$ of exceptional times at which a non-trivial bi-infinite geodesic exists a.s. has Hausdorff dimension at most $1/2$. Further, for any fixed direction $\theta$, we show that the set $\mathscr{T}^\theta\subseteq \mathscr{T}$ of times at which a non-trivial bi-infinite geodesic directed along $\theta$ exists a.s. has Hausdorff dimension equal to $0$.