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 $\varepsilon&gt;0$, the union of all geodesics between two KPZ-scale rectangles of transverse width of order $n<sup>{2/3}$ and longitudinal length of order $n$, separated by a distance of order $n$, visits at most $n<sup>{1+\varepsilon}$ unit horizontal cells in the bulk during the critical time interval $[0,n<sup>{-1/3}]$, both in expectation and with stretched-exponentially high probability. We also obtain the quantitative bound $n\exp{C(\log\log n)<sup>2}$ on the expected hitset size, with a corresponding failure probability at most $Ce^{-c(\log n)<sup>2}$. 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.