Large deviations in the geometry of convex lattice polygons
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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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- Large deviations in the geometry of convex lattice polygons
Israel Journal of Mathematics
Volume 109, Issue 1 , pp 13-27
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- 1. St. Petersburg Branch, Mathematical Institute, Russian Academy of Sciences, Fontanka 27, 191011, St. Petersburg, Russia
- 2. Institute for Advanced Studies, The Hebrew University of Jerusalem, 91904, Jerusalem, Israel
- 3. Department of Electrical Engineering, Technion-Israel Institute of Technology, 32000, Haifa, Israel