Supremum of the Airy2 Process Minus a Parabola on a Half Line

@article{Quastel2013SupremumOT,
  title={Supremum of the Airy2 Process Minus a Parabola on a Half Line},
  author={Jeremy Quastel and Daniel Remenik},
  journal={Journal of Statistical Physics},
  year={2013},
  volume={150},
  pages={442-456}
}
AbstractLet $\mathcal {A}_{2}(t)$ be the Airy2 process. We show that the random variable $$\sup_{t\leq\alpha} \bigl\{\mathcal {A}_2(t)-t^2 \bigr\}+\min\{0,\alpha \}^2 $$ has the same distribution as the one-point marginal of the Airy2→1 process at time α. These marginals form a family of distributions crossing over from the GUE Tracy-Widom distribution FGUE(x) for the Gaussian Unitary Ensemble of random matrices, to a rescaled version of the GOE Tracy-Widom distribution FGOE(41/3x) for the… 
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