Exceedingly Large Deviations of the Totally Asymmetric Exclusion Process

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.