  Consider the Totally Asymmetric Simple Exclusion Process (TASEP) on the
integer lattice $ \mathbb{Z} $. We study the functional Large Deviations of the
integrated current $ \mathsf{h}(t,x) $ under the hyperbolic scaling of space
and time by $ N $, i.e., $ \mathsf{h}_N(t,\xi) :=
\frac{1}{N}\mathsf{h}(Nt,N\xi) $. As hinted by the asymmetry in the upper- and
lower-tail large deviations of the exponential Last Passage Percolation, the
TASEP exhibits two types of deviations. One type of deviations occur with
probability $ \exp(-O(N)) $, referred to as speed-$ N $; while the other with
probability $ \exp(-O(N^{2})) $, referred to as speed-$ N^2 $. In this work we
study the speed-$ N^2 $ functional LDP of the TASEP, and establishes
(non-matching) large deviation upper and lower bounds.
