Exceedingly Large Deviations of the Totally Asymmetric Exclusion Process

TL;DR

This paper investigates the large deviation principles for the integrated current in TASEP under hyperbolic scaling, revealing two distinct deviation regimes with different exponential decay rates.

Contribution

It establishes the first large deviation upper and lower bounds at the speed-$N^2$ scale for TASEP's integrated current, highlighting the complexity of its deviation behavior.

Findings

Identifies two types of large deviations with different probabilities.

Provides non-matching upper and lower bounds for speed-$N^2$ deviations.

Enhances understanding of rare events in TASEP dynamics.

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 LDP of the TASEP, and establishes (non-matching) large deviation upper and lower bounds.