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Continuum Statistics of the Airy2 Process

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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 \({\sup(\mathcal{A}_{2}(x)-x^2)}\) 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 (Commun. Math. Phys. 242(1–2):277–329, 2003) observation that the GOE result follows from this polymer interpretation and exact results within that field. In a companion paper (Moreno Flores et al. in Commun. Math. Phys. 2012) these continuum statistics are used to compute the distribution of the endpoint of directed polymers.

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Authors and Affiliations

  1. Courant Institute, 251 Mercer St., Room 604, New York, NY, 10012, USA

    Ivan Corwin

  2. Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, Ontario, M5S 2E4, Canada

    Jeremy Quastel & Daniel Remenik

  3. Departamento de Ingeniería Matemática, Universidad de Chile, Av. Blanco Encalada 2120, Santiago, Chile

    Daniel Remenik

Authors
  1. Ivan Corwin
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  2. Jeremy Quastel
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  3. Daniel Remenik
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Correspondence to Jeremy Quastel.

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Communicated by H.-T. Yau

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Corwin, I., Quastel, J. & Remenik, D. Continuum Statistics of the Airy2 Process. Commun. Math. Phys. 317, 347–362 (2013). https://doi.org/10.1007/s00220-012-1582-0

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