  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).
