The lower tail of the half-space KPZ equation

Abstract

We establish the first tight bound on the lower tail probability of the half-space KPZ equation with Neumann boundary parameter A=−1/2 and narrow-wedge initial data. The lower bound demonstrates a crossover between two regimes of super-exponential decay with exponents 52 and 3; the upper bound demonstrates a crossover between regimes with exponents 32 and 3. Given a crude leading-order asymptotic in the Stokes region for the Ablowitz–Segur solution to Painlevé II (Definition 1.8), we improve the upper bound to demonstrate the same crossover as the lower bound. We also establish novel bounds on the large deviations of the GOE point process.