Connecting Eigenvalue Rigidity with Polymer Geometry: Diffusive Transversal Fluctuations under Large Deviation

We consider the exactly solvable model of exponential directed last passage percolation on $\mathbb{Z}^2$ in the large deviation regime. Conditional on the upper tail large deviation event $\mathcal{U}_{\delta}:=\{T_{n}\geq (4+\delta)n\}$ where $T_{n}$ denotes the last passage time from $(1,1)$ to $(n,n)$, we study the geometry of the polymer/geodesic $\Gamma_{n}$, i.e., the optimal path attaining $T_{n}$. We show that conditioning on $\mathcal{U}_{\delta}$ changes the transversal fluctuation exponent from the characteristic $2/3$ of the KPZ universality class to $1/2$, i.e., conditionally, the smallest strip around the diagonal that contains $\Gamma_{n}$ has width $n^{1/2+o(1)}$ with high probability. This sharpens a result of Deuschel and Zeitouni (1999) who proved a $o(n)$ bound on the transversal fluctuation in the context of Poissonian last passage percolation, and complements (Basu, Ganguly, Sly, 2017), where the transversal fluctuation was shown to be $\Theta(n)$ in the lower tail large deviation event. Our proof exploits the correspondence between last passage times in the exponential LPP model and the largest eigenvalue of the Laguerre Unitary Ensemble (LUE) together with the determinantal structure of the spectrum of the latter. A key ingredient in our proof is a sharp refinement of the large deviation result for the largest eigenvalue (Sepp\"al\"ainen '98, Johansson '99), using rigidity properties of the spectrum, which could be of independent interest.