Geodesic switches and exceptional times in dynamical Brownian last passage percolation

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 $0$.