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Limit shapes of typical geometric configurations and their applications. (English. Russian original) Zbl 1146.51303

J. Math. Sci., New York 119, No. 2, 165-177 (2004); translation from Zap. Nauchn. Semin. POMI 280, 73-100 (2001).
This is the text of an expository lecture to students of the Independent University of Moscow, describing recent work, mainly of the author (together with I. Bárány in the case of lattice polygons [Geom. Funct. Anal. 2, No. 4, 381–393 (1992; Zbl 0772.52010)], and with Yu. Yakubovich in the case of Young tableaux [Mosc. Math. J. 1, No. 3, 457–468 (2001; Zbl 0996.05006)]), on the “limit shapes” of geometric objects (in particular, lattice polygons up to affine equivalence) and Young tableaux.
The author presents a fascinating journey through this area touching upon diverse topics such as combinatorics, probability theory, affine differential geometry, statistical physics, variational principles, asymptotic analysis, crystallography etc.

MSC:

51M15 Geometric constructions in real or complex geometry
05A17 Combinatorial aspects of partitions of integers
11Z05 Miscellaneous applications of number theory
11P82 Analytic theory of partitions
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
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