Geodesic traces in dynamical Brownian last passage percolation

Abstract

We consider Brownian last passage percolation (BLPP) in which the Brownian increment process on each unit horizontal interval is independently resampled at rate one. By combining strong passage-time stability estimates with a multiscale analysis of static near-optimal paths, we show that, for every $\varepsilon>0$, the union of all geodesics between two KPZ-scale rectangles of transverse width of order $n^{2/3}$ and longitudinal length of order $n$, separated by a distance of order $n$, visits at most $n^{1+\varepsilon}$ unit horizontal cells in the bulk during the critical time interval $[0,n^{-1/3}]$, both in expectation and with stretched-exponentially high probability. We also obtain the quantitative bound $n\exp\{C(\log\log n)^2\}$ on the expected hitset size, with a corresponding failure probability at most $Ce^{-c(\log n)^2}$. Using this, we establish that the set of times admitting a non-trivial bigeodesic has almost surely zero Hausdorff measure for the subpolynomially decaying gauge $H(r)=\exp\{-L(r)^2\log L(r)\}$, where $L(r)=\log\log(1/r)$, as $r\downarrow0$. In particular, this set almost surely has Hausdorff dimension zero. For each fixed deterministic non-axial direction, we further show that almost surely no time admits a bigeodesic in that direction.