Israel Journal of Mathematics

, Volume 109, Issue 1, pp 13–27

Large deviations in the geometry of convex lattice polygons


DOI: 10.1007/BF02775023

Cite this article as:
Vershik, A. & Zeitouni, O. Isr. J. Math. (1999) 109: 13. doi:10.1007/BF02775023


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.

Copyright information

© The Magnes Press 1999

Authors and Affiliations

  1. 1.St. Petersburg Branch, Mathematical InstituteRussian Academy of SciencesSt. PetersburgRussia
  2. 2.Institute for Advanced StudiesThe Hebrew University of JerusalemJerusalemIsrael
  3. 3.Department of Electrical EngineeringTechnion-Israel Institute of TechnologyHaifaIsrael