Continuum statistics of the Airy2 process

Abstract

We develop an exact determinantal formula for the probability that the Airy${}_2$ process is bounded by a function $g$ on a finite interval. As an application, we provide a direct proof that $\sup(\aip(x)-x^2)$ is distributed as a GOE random variable. Both the continuum formula and the GOE result have applications in the study of the end point of an unconstrained directed polymer in a disordered environment. We explain Johansson's [Joh03] observation that the GOE result follows from this polymer interpretation and exact results within that field. In a companion paper [MQR11] these continuum statistics are used to compute the distribution of the endpoint of directed polymers.

More details added, some minor mistakes corrected

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• Local behavior and hitting probabilities of the Airy1 process
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• Tails of the endpoint distribution of directed polymers
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• Supremum of the Airy2 process minus a parabola on a half line
• Kardar-Parisi-Zhang universality class for the critical dynamics of reaction-diffusion fronts
• Brownian Gibbs property for Airy line ensembles
• The hard-edge tacnode process for Brownian motion
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• Kardar-Parisi-Zhang Universality in First-Passage Percolation: the Role of Geodesic Degeneracy
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• Shape effects in the fluctuations of random isochrones on a square lattice
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• Painleve II in random matrix theory and related fields
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• On the average of the Airy process and its time reversal
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• Ergodicity of the Airy line ensemble