Connecting eigenvalue rigidity with polymer geometry: Diffusive transversal fluctuations under large deviation

We consider exponential directed last passage percolation (LPP) on Z2, a paradigm model of the Kardar-Parisi-Zhang (KPZ) universality class, where Tn denotes the last passage time from (1, 1) to (n, n), and I'n denotes the corresponding polymer, i.e., the optimal path attaining Tn. The typical fluctuation of the geodesic from the straight line joining its endpoints is known to be of order n2/3, a feature of KPZ universality. Despite considerable interest, the behaviour of the polymer under large deviation events for Tn had remained less understood. In this paper we consider the upper tail large deviation event U8 :_ {Tn >= (4 + 8)n}. We show that conditioning on U8 changes the transversal fluctuation exponent from 2/3 to 1/2, i.e., conditionally, the smallest strip around the diagonal that contains I'n has width n1/2+o(1) with high probability. While earlier work by Deuschel and Zeitouni (Combin. Probab. Comput. 8 (1999) 247-263) had a o(n) upper bound for the transversal fluctuation conditional on the upper tail large deviations in Poissonian last passage percolation, the exponent 1/2 is new and is expected to be universal across various planar last passage percolation models in the KPZ universality class. Our proof combines several different ideas exploiting the correspondence between last passage times in the exponential LPP model and the largest eigenvalue of the Laguerre Unitary Ensemble (LUE), including a stochastic monotonicity result for determinantal point processes, as well as recent advances in understanding rigidity properties of eigenvalues to obtain a sharp finite size correction to the well known large deviation rate function for the largest eigenvalue.