Exact decay of the persistence probability in the Airy1 process

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Abstract
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.

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