Bogachev, LV and Zarbaliev, SM (2011) Universality of the limit shape of convex lattice polygonal lines. Annals of Probability, 39 (6). 2271  2317 (47). ISSN 00911798

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Abstract
Let Π_n be the set of convex polygonal lines Γ with vertices on Z^2_+ and fixed endpoints 0 = (0, 0) and n = (n_1, n_2). We are concerned with the limit shape, as n → ∞, of "typical" Γ ∈ Π_n with respect to a parametric family of probability measures {P^r_n, 0 < r < 1} on Π_n, 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 {P^r_n}, even though P^r_n (r ≠ 1) and P^1_n are asymptotically singular. Measures P^r_n are constructed, following Sinai’s approach, as conditional distributions Q^r_z(·Π_n), where Q^r_z are suitable product measures on the space Π = U_n Π_n, depending on an auxiliary "free" parameter z = (z1,z2). The transition from (Π,Q^r_z) to (Π_n,P^r_n) is based on the asymptotics of the probability Q^r_z(Π_n), furnished by a certain twodimensional local limit theorem. The proofs involve subtle analytical tools including the Möbius inversion formula and properties of zeroes of the Riemann zeta function.
Item Type:  Article 

Copyright, Publisher and Additional Information:  © 2011 Institute of Mathematical Statistics. Reproduced in accordance with the publisher's selfarchiving policy. 
Keywords:  Convex lattice polygonal lines; Limit shape; Randomization; Local limit theorem 
Institution:  The University of Leeds 
Academic Units:  The University of Leeds > Faculty of Maths and Physical Sciences (Leeds) > School of Mathematics (Leeds) > Statistics (Leeds) 
Depositing User:  Symplectic Publications 
Date Deposited:  13 Oct 2011 13:52 
Last Modified:  04 Dec 2014 12:07 
Published Version:  http://dx.dio.org/10.1214/10AOP607 
Status:  Published 
Publisher:  Institute of Mathematical Statistics 
Refereed:  Yes 
Identification Number:  10.1214/10AOP607 
URI:  http://eprints.whiterose.ac.uk/id/eprint/43328 