Lyapunov Exponents of the Half-Line SHE

Upper-tail large deviation principle for the ASEP

“…This leads to the proof of Theorem 1.1. The above description of our method is in line with the Lyapunov moment approach adopted in the works of [DT19], [GL20] and [Lin20] to obtain upper-tail large deviation results of other integrable models, such as the KPZ equation. Namely, we extract fractional moments from the (τ -)Laplace transform such as (1.7) according to Lemma 1.8.…”

“…Indeed, K ζ,t is asymmetric and as u varies in (δ − i∞, δ + i∞), the function gt (w) gt(τ u w) appearing in the kernel K ζ,t , exhibits a periodic behavior, whereas the kernel in the KPZ models involves Airy functions in its integrand which have a unique maximum and are much easier to analyze. Furthermore, our model exhibits exponentially decaying moments of τ H 0 (t) as opposed to the exponentially increasing ones of the KPZ models in [DT19] and [Lin20] and this demands a more precise understanding of the trace term of our Fredholm determinant expansion. For instance in Section 3, to obtain the precise asymptotics for our leading term, we have performed steepest descent analysis on the kernel K ζ,t , where the periodic nature of gt(w) gt(τ u w) results in infinitely many critical points.…”

“…Namely, we extract fractional moments from the (τ -)Laplace transform such as (1.7) according to Lemma 1.8. In particular, our work draws from those of [DT19] and [Lin20], which studied the fractional moments of the Stochastic Heat Equation (SHE) and the half-line Stochastic Heat Equation, respectively. We will further contextualize the connections of our work to [DT19], [GL20] and [Lin20] in Section 1.3.…”

“…In particular, our work draws from those of [DT19] and [Lin20], which studied the fractional moments of the Stochastic Heat Equation (SHE) and the half-line Stochastic Heat Equation, respectively. We will further contextualize the connections of our work to [DT19], [GL20] and [Lin20] in Section 1.3. In the following text, however, we emphasize a few key differences and technical challenges unique to the ASEP that we have encountered and resolved in our proof.…”

“…A major technical challenge in our proof is to argue how the contribution from only one of the critical points dominates the those from the rest and this is accomplished in the proof of Proposition 2.4. Similarly, the asymmetry of the kernel in the ASEP model has led us to opt for the Hadamard's inequality approach as exemplified in Section 4 of [Lin20], instead of the operator theory argument in [DT19], to obtain a sufficient upper bound for the higher-order terms in our paper in Section 4.…”