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November 2011 Universality of the limit shape of convex lattice polygonal lines
Leonid V. Bogachev, Sakhavat M. Zarbaliev
Ann. Probab. 39(6): 2271-2317 (November 2011). DOI: 10.1214/10-AOP607

Abstract

Let Πn be the set of convex polygonal lines Γ with vertices on ℤ+2 and fixed endpoints 0 = (0, 0) and n = (n1, n2). We are concerned with the limit shape, as n → ∞, of “typical” ΓΠn with respect to a parametric family of probability measures {Pnr, 0 < r < ∞} on Πn, including the uniform distribution (r = 1) for which the limit shape was found in the early 1990s independently by A. M. Vershik, I. Bárány and Ya. G. Sinai. We show that, in fact, the limit shape is universal in the class {Pnr}, even though Pnr (r ≠ 1) and Pn1 are asymptotically singular. Measures Pnr are constructed, following Sinai’s approach, as conditional distributions Qzr(⋅|Πn), where Qzr are suitable product measures on the space Π = ⋃nΠn, depending on an auxiliary “free” parameter z = (z1, z2). The transition from (Π, Qzr) to (Πn, Pnr) is based on the asymptotics of the probability Qzr(Πn), furnished by a certain two-dimensional local limit theorem. The proofs involve subtle analytical tools including the Möbius inversion formula and properties of zeroes of the Riemann zeta function.

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Leonid V. Bogachev. Sakhavat M. Zarbaliev. "Universality of the limit shape of convex lattice polygonal lines." Ann. Probab. 39 (6) 2271 - 2317, November 2011. https://doi.org/10.1214/10-AOP607

Information

Published: November 2011
First available in Project Euclid: 17 November 2011

zbMATH: 1242.52007
MathSciNet: MR2932669
Digital Object Identifier: 10.1214/10-AOP607

Subjects:
Primary: 05A17 , 52A22
Secondary: 05D40 , 60F05

Keywords: Convex lattice polygonal lines , limit shape , local limit theorem , Randomization

Rights: Copyright © 2011 Institute of Mathematical Statistics

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Vol.39 • No. 6 • November 2011
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