An upper bound on geodesic length in 2D critical first-passage percolation

We consider i.i.d. first-passage percolation (FPP) on the two-dimensional square lattice, in the critical case where edge-weights take the value zero with probability $\tfrac{1}{2}$. Critical FPP is unique in that the Euclidean lengths of geodesics are superlinear -- rather than linear -- in the distance between their endpoints. This fact was speculated by Kesten in 1986 but not confirmed until 2019 by Damron and Tang, who showed a lower bound on geodesic length that is polynomial with degree strictly greater than $1$. In this paper, we establish the first nontrivial upper bound. Namely, we prove that for a large class of critical edge-weight distributions, the shortest geodesic from the origin to a box of radius $R$ uses at most $R^{2+ε}π_3(R)$ edges with high probability, for any $ε> 0$. Here $π_3(R)$ is the polychromatic 3-arm probability from classical Bernoulli percolation; upon inserting its conjectural asymptotic, our bound converts to $R^{4/3 + ε}$. In any case, it is known that $π_3(R) \lesssim R^{-δ}$ for some $δ> 0$, so our bound gives an exponent strictly less than $2$. In the special case of Bernoulli($\tfrac{1}{2}$) edge-weights, we replace the additional factor of $R^ε$ with a constant and give an expectation bound.