Exact decay of the persistence probability in the Airy₁ process

We consider the Airy₁ process, which is the limit process in KPZ growth models with flat and non-random initial conditions. We study the persistence probability, namely the probability that the process stays below a given threshold c for a time span of length L. This is expected to decay as e-κ(c) L. We determine an analytic expression for κ(c) for all c≥ 3/2 starting with the continuum statistics formula for the persistence probability. As the formula is analytic only for $c>0$, we determine an analytic continuation of κ(c) and numerically verify the validity for $c<0$ as well.