Exceedingly large deviations of the totally asymmetric exclusion process

Abstract

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 Large Deviation Principle (LDP) of the TASEP, and establish (non-matching) large deviation upper and lower bounds.