Exceedingly large deviations of the totally asymmetric exclusion process

Consider the Totally Asymmetric Simple Exclusion Process (TASEP) on the integer lattice Z. We study the functional Large Deviations of the integrated current h(t, x) under the hyperbolic scaling of space and time by N, i. e., h(N)(t, xi) := 1/Nh(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 Large Deviation Principle (LDP) of the TASEP, and establish (non-matching) large deviation upper and lower bounds.