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The Pearcey Process

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Abstract

The extended Airy kernel describes the space-time correlation functions for the Airy process, which is the limiting process for a polynuclear growth model. The Airy functions themselves are given by integrals in which the exponents have a cubic singularity, arising from the coalescence of two saddle points in an asymptotic analysis. Pearcey functions are given by integrals in which the exponents have a quartic singularity, arising from the coalescence of three saddle points. A corresponding Pearcey kernel appears in a random matrix model and a Brownian motion model for a fixed time. This paper derives an extended Pearcey kernel by scaling the Brownian motion model at several times, and a system of partial differential equations whose solution determines associated distribution functions. We expect there to be a limiting nonstationary process consisting of infinitely many paths, which we call the Pearcey process, whose space-time correlation functions are expressible in terms of this extended kernel.

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Authors and Affiliations

  1. Department of Mathematics, University of California, Davis, CA, 95616, USA

    Craig A. Tracy

  2. Department of Mathematics, University of California, Santa Cruz, CA, 95064, USA

    Harold Widom

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  1. Craig A. Tracy
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  2. Harold Widom
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Communicated by H. Spohn

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Tracy, C., Widom, H. The Pearcey Process. Commun. Math. Phys. 263, 381–400 (2006). https://doi.org/10.1007/s00220-005-1506-3

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  • DOI: https://doi.org/10.1007/s00220-005-1506-3

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