The lower tail of the half-space KPZ equation

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.