The lower tail of the half-space KPZ equation

We establish the first tight bounds on the lower tail probability of the half-space KPZ equation with Neumann boundary parameter $A = -1/2$ and narrow-wedge initial data at the boundary point. These bounds hold for all sufficiently large $T$ and demonstrate a crossover when the depth is approximately of order $T^{2/3}$ 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 $\frac{5}{2}$ exponent and its pre-factor was first observed in [KLD18b]; the cubic exponent and its pre-factor is indicative of the limiting tail-decay following the GOE Tracy-Widom distribution.