Vershik, A.; Zeitouni, O. Large deviations in the geometry of convex lattice polygons. (English) Zbl 0945.60022 Isr. J. Math. 109, 13-27 (1999). From the authors’ 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-Minkowski transform of convex curves plays a crucial role in our approach. Reviewer: S.Poghosyan (Erevan) Cited in 1 ReviewCited in 7 Documents MSC: 60F10 Large deviations Keywords:large deviations; Gauss-Minkowski transform; convex polygons PDFBibTeX XMLCite \textit{A. Vershik} and \textit{O. Zeitouni}, Isr. J. Math. 109, 13--27 (1999; Zbl 0945.60022) Full Text: DOI References: [1] Bárány, I., The limit shape of convex lattice polygons, Discrete and Computational Geometry, 13, 279-295 (1995) · Zbl 0824.52001 [2] Bárány, I., Affine perimeter and limit shape, Journal für die reine und angewandte Mathematik, 484, 71-84 (1997) · Zbl 0864.52010 [3] Buchin, Su, Affine Differential Geometry (1983), London: Gordon and Breach, London · Zbl 0539.53002 [4] Buseman, H., Convex Surfaces (1958), New York: Interscience Publ., New York · Zbl 0196.55101 [5] Dembo, A.; Zeitouni, O., Large deviations techniques and applications (1993), New York: Springer, New York · Zbl 0793.60030 [6] Deuschel, J. D.; Zeitouni, O., Limiting curves for i.i.d. records, The Annals of Probability, 23, 852-878 (1995) · Zbl 0834.60058 [7] R. Schneider,Convex Bodies: The Brunn-Minkowskii Theory, Cambridge University Press, 1993. · Zbl 0798.52001 [8] Sinai, Ya. G., The probabilistic approach to the analysis of statistics for convex polygonal lines, Functional Analysis and its Applications, 28, 108-113 (1994) · Zbl 0832.60099 [9] Vershik, A. M., The limit shape of convex lattice polygons and related topics, Functional Analysis and its Applications, 28, 13-20 (1994) · Zbl 0848.52004 [10] Vershik, A. M., Statistical mechanics of combinatorial partitions and their limit shapes, Functional Analysis and its Applications, 30, 19-30 (1996) · Zbl 0868.05004 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.