UNIVERSALITY OF THE LIMIT SHAPE OF CONVEX LATTICE POLYGONAL LINES

• 成果类型: Article
• 署名作者: Bogachev, Leonid V.; Zarbaliev, Sakhavat M.
• 署名单位: University of Leeds; Russian Academy of Sciences; Institute of Earthquake Prediction Theory & Mathematical Geophysics
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/10-AOP607
• 发表日期: 2011
• 页码: 2271-2317
• 关键词: partitions

摘要

Let Pi(n) be the set of convex polygonal lines Gamma 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 -> 8, of typical Gamma is an element of Pi(n) with respect to a parametric family of probability measures {P-n(r), 0 < r < infinity} on Pi(n), including the uniform distribution (r = 1) for which the limit shape was found in the early 1990s independently by A. M. Vershik, I. Barany and Ya. G. Sinai. We show that, in fact, the limit shape is universal in the class {P-n(r)}, even though P-n(r) (r not equal 1) and P-n(1) are asymptotically singular. Measures P-n(r) are constructed, following Sinai's approach, as conditional distributions Q(z)(r)(center dot vertical bar Pi(n)), where Q(z)(r) are suitable product measures on the space Pi = boolean OR(n) Pi(n) Pi(n), depending on an auxiliary free parameter z = (z(1), z(2)). The transition from (Pi, Q(z)(r)) to (Pi(n), P-n(r)) is based on the asymptotics of the probability Q(z)(r)(Pi(n)), furnished by a certain two-dimensional local limit theorem. The proofs involve subtle analytical tools including the Mobius inversion formula and properties of zeroes of the Riemann zeta function.