Linear-order dynamical hitsets with uniform tails

Determine whether the expected dynamical hitset for Brownian last passage percolation, between the endpoint regions \(\mathscr R_n^-\) and \(\mathscr R_n^+\) and over the critical time interval \([0,n^{-1/3}]\) in the bulk slab \([-n/2,n/2]\), is bounded by \(Cn\) for an absolute constant \(C\); additionally, establish whether there are constants \(C,c>0\) and \(\gamma\in(0,1)\), independent of \(n\) and \(\alpha\), such that the corresponding hitset exceeds \(\alpha n\) with probability at most \(Ce^{-c\alpha^\gamma}\) for every integer \(n\ge1\) and \(\alpha\ge1\).

Background

Theorem 1 establishes an expected hitset bound of order $n\exp\{C(\log\log n)^2\}$ on the critical dynamical time interval, together with a stretched-exponential failure estimate at a comparable threshold. The question asks whether the subpolynomial correction can be removed, yielding a genuinely linear expected hitset size, and whether a uniform tail bound in the excess factor $\alpha$ holds.

A positive answer to the expectation bound would substantially strengthen the control of geodesic traces and, as discussed immediately afterward, is expected to imply almost-sure nonexistence of exceptional times supporting bigeodesics with random directions. The authors explain that their current multiscale argument incurs scale-dependent branching and terminal costs, and does not provide the required uniform-in-$n$ tail estimate.