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Polynuclear Growth on a Flat Substrate and Edge Scaling of GOE Eigenvalues

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Abstract

We consider the polynuclear growth (PNG) model in 1+1 dimension with flat initial condition and no extra constraints. Through the Robinson-Schensted-Knuth (RSK) construction, one obtains the multilayer PNG model, which consists of a stack of non-intersecting lines, the top one being the PNG height. The statistics of the lines is translation invariant and at a fixed position the lines define a point process. We prove that for large times the edge of this point process, suitably scaled, has a limit. This limit is a Pfaffian point process and identical to the one obtained from the edge scaling of the Gaussian orthogonal ensemble (GOE) of random matrices. Our results give further insight to the universality structure within the KPZ class of 1+1 dimensional growth models.

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Authors and Affiliations

  1. Zentrum Mathematik, Technische Universität München, Boltzmannstr. 3, 85747, Garching bei München, Germany

    Patrik L. Ferrari

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  1. Patrik L. Ferrari
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Correspondence to Patrik L. Ferrari.

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Communicated by M. Aizenman

To Freeman Dyson on the occasion of his eightieth birthday

Acknowledgement The author would like to thank Michael Prähofer and Herbert Spohn for discussions about the present work, Tomohiro Sasamoto for explanations on the growth model in half-space, Kurt Johansson for suggesting the problem, Jani Lukkarinen for discussions on technical questions, and József Lőrinczi for reading part of the manuscript. Thanks go also to the referees for critical reading and useful suggestions.

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Ferrari, P. Polynuclear Growth on a Flat Substrate and Edge Scaling of GOE Eigenvalues. Commun. Math. Phys. 252, 77–109 (2004). https://doi.org/10.1007/s00220-004-1204-6

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