  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.
