The Annals of Probability

Universality of the limit shape of convex lattice polygonal lines

Leonid V. Bogachev and Sakhavat M. Zarbaliev

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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.

Article information

Source
Ann. Probab., Volume 39, Number 6 (2011), 2271-2317.

Dates
First available in Project Euclid: 17 November 2011

Permanent link to this document
https://projecteuclid.org/euclid.aop/1321539121

Digital Object Identifier
doi:10.1214/10-AOP607

Mathematical Reviews number (MathSciNet)
MR2932669

Zentralblatt MATH identifier
1242.52007

Subjects
Primary: 52A22: Random convex sets and integral geometry [See also 53C65, 60D05] 05A17: Partitions of integers [See also 11P81, 11P82, 11P83]
Secondary: 60F05: Central limit and other weak theorems 05D40: Probabilistic methods

Keywords
Convex lattice polygonal lines limit shape randomization local limit theorem

Citation

Bogachev, Leonid V.; Zarbaliev, Sakhavat M. Universality of the limit shape of convex lattice polygonal lines. Ann. Probab. 39 (2011), no. 6, 2271--2317. doi:10.1214/10-AOP607. https://projecteuclid.org/euclid.aop/1321539121


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