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 \(Ce^{-c{t}^3}\) . In this note, we provide a short argument establishing a matching lower bound for this probability.
