Determine the precise polynomial prefactor for branching-random-walk last passage time

Determine the precise exponent governing the polynomial-in-[?]n prefactor of the last passage time for the branching random walk, beyond the established lower and upper bounds.

Background

The paper compares last passage percolation on fractal percolation with last passage percolation driven by the branching random walk, a hierarchical approximation to the two-dimensional Gaussian free field. The cited results provide a lower bound of order 2n n{1/2}/log n and an upper bound of order 2n n. The authors identify the remaining uncertainty as the precise polynomial factor in n separating these bounds.

References

While the precise exponent governing the $\mathrm{poly}(n)$ pre-factor remains unknown, it is not unreasonable to speculate that the lower bound is sharp, for reasons that will become clearer through the course of this discussion.

— Directed fractal percolation and non-Lipschitz variants  (2610.12366 - Ganguly et al., 8 Oct 2026) in Section 1, subsection “LPP on the Gaussian free field (GFF)”

In fact, it is not unnatural to extrapolate from the proof strategy for Theorem \ref{t:sublinear} to conjecture that for any $\gamma>1/2$, the maximum number of points of value $n{\gamma}$ on any directed path is at most $2n e{- n{\beta}}$ for some $\beta=\beta(\gamma)>0$, which would then imply the sharpness of the lower bound in lb342 up to sub-$\mathrm{poly}(n)$ factors.

lb342:

LnBRW2nn1/2log⁡n.L_n^{{BRW}} \frac{2^n n^{1/2}}{\log n}.

— Directed fractal percolation and non-Lipschitz variants  (2610.12366 - Ganguly et al., 8 Oct 2026) in Section 1, subsection “LPP on the Gaussian free field (GFF)”