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Large deviations in the geometry of convex lattice polygons

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Abstract

We provide a full large deviation principle (LDP) for the uniform measure on certain ensembles of convex lattice polygons. This LDP provides for the analysis of concentration of the measure on convex closed curves. In particular, convergence to a limiting shape results in some particular cases, including convergence to a circle when the ensemble is defined as those centered convex polygons, with vertices on a scaled two dimensional lattice, and with length bounded by a constant. The Gauss-Minkowskii transform of convex curves plays a crucial role in our approach.

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Authors and Affiliations

  1. St. Petersburg Branch, Mathematical Institute, Russian Academy of Sciences, Fontanka 27, 191011, St. Petersburg, Russia

    A. Vershik

  2. Institute for Advanced Studies, The Hebrew University of Jerusalem, 91904, Jerusalem, Israel

    A. Vershik

  3. Department of Electrical Engineering, Technion-Israel Institute of Technology, 32000, Haifa, Israel

    O. Zeitouni

Authors
  1. A. Vershik
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  2. O. Zeitouni
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Corresponding author

Correspondence to A. Vershik.

Additional information

Partially supported by grants ININS 94-3420 and RFF1-96-01-00676.

Partially supported by a US-Israel BSF grant and by the fund for promotion of research at the Technion.

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Vershik, A., Zeitouni, O. Large deviations in the geometry of convex lattice polygons. Isr. J. Math. 109, 13–27 (1999). https://doi.org/10.1007/BF02775023

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  • DOI: https://doi.org/10.1007/BF02775023

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