Lower bound for large local transversal fluctuations of Geodesics in Last Passage Percolation

For exactly solvable models of planar last passage percolation, it is known that geodesics of length $n$ exhibit transversal fluctuations at scale $n^{2/3}$ and matching (up to exponents) upper and lower bounds for the tail probabilities are available. The local transversal fluctuations near the endpoints are expected to be much smaller; it is known that the transversal fluctuation up to distance $r \ll n$ is typically of the order $r^{2/3}$ and the probability that the fluctuation is larger than $tr^{2/3}$ is at most $Ce^{-ct^3}$. In this note, we provide a short argument establishing a matching lower bound for this probability.