Lower bound for large midpoint transversal fluctuations in the corner growth model

The study of transversal fluctuation of the optimal path has been a crucial aspect of the KPZ universality class. In this paper, we establish the first probability lower bound, with optimal exponential order, for the rare event in which the midpoint (or any fixed level) of the geodesic has a large transversal fluctuation. We present our result in the setting of the corner growth model, which is also known as the exponential last-passage percolation. The previously known lower bounds hold for the maximum transversal fluctuation along the entire geodesic (Hammond-Sarkar'20) or the starting portion of the geodesic at a local scale (Agarwal'23). Our result improves upon these as now the rare event can demand where the large fluctuation occurs exactly along the geodesic, on both local and global scales. This fills the final missing piece in the literature on the tail bounds for the transversal fluctuation of geodesics. Our proof utilizes the coupling method: we first obtain a version of the estimate for the semi-infinite geodesic using duality and then transfer the result to finite geodesics using path monotonicity. Our method differs from the previous works (Agarwal'23, Hammond-Sarkar'20), and in fact, we do not require fine information about the left tail moderate deviation of the last-passage time, which played a crucial role in (Agarwal'23, Hammond-Sarkar'20).