Lower bound for large transversal fluctuations in exactly solvable KPZ models

The study of transversal fluctuation of the optimal path has been a crucial aspect of the Kadar-Parisi-Zhang (KPZ) universality class. In this paper, we establish a new probability lower bound, with optimal exponential order, for the rare event in which a given level of the optimal path has a large transversal fluctuation. We present our results in both zero and positive temperature settings. The previously known lower bounds were obtained in zero temperature models, and they 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 optimal path, on both local and global scales. Our proof utilizes the coupling method: we first obtain a version of the estimate in the semi-infinite setting using duality and then transfer the result to finite paths using planar 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, which played a crucial role in (Agarwal'23, Hammond-Sarkar'20).