# The lower tail of the half-space KPZ equation

## TL;DR

This paper provides the first tight bounds on the lower tail probability of the half-space KPZ equation with specific boundary conditions, revealing a crossover in decay regimes and connecting to Painlevé II asymptotics.

## Contribution

It establishes the first tight bounds on the lower tail of the half-space KPZ equation, identifying crossover regimes and linking bounds to Painlevé II asymptotics.

## Key findings

- Demonstrates a crossover between super-exponential decay regimes with different exponents.
- Shows the upper bound can be improved to match the lower bound crossover.
- Provides new bounds on large deviations of the GOE point process.

## 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. When the tail depth is of order $T^{2/3}$, the lower bound demonstrates a crossover between a regime of super-exponential decay with exponent $\frac{5}{2}$ (and leading pre-factor $\frac{2}{15 \pi}T^{1/3}$) and a regime with exponent $3$ (and leading pre-factor $\frac{1}{24}$); the upper bound demonstrates a crossover between a regime with exponent $\frac{3}{2}$ (and arbitrarily small pre-factor) and a regime with exponent $3$ (and leading pre-factor $\frac{1}{24}$). We show that, given a crude leading-order asymptotic in the Stokes region (Definition $1.7$, first defined in (Duke Math J., [Bot17])) for the Ablowitz-Segur solution to the Painlev\'e II equation, the upper bound on the lower tail probability can be improved to demonstrate the same crossover as the lower bound. We also establish novel bounds on the large deviations of the GOE point process.