Universality of the limit shape of convex lattice polygonal lines

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 = (n₁, n₂). We are concerned with the limit shape, as n → ∞, of “typical” Γ ∈ Π_n with respect to a parametric family of probability measures {P_n^r, 0 < r < ∞} 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_n^r}, even though P_n^r (r ≠ 1) and P_n¹ are asymptotically singular. Measures P_n^r are constructed, following Sinai’s approach, as conditional distributions Q_z^r(⋅|Π_n), where Q_z^r are suitable product measures on the space Π = ⋃_n Π_n, depending on an auxiliary “free” parameter z = (z₁, z₂). The transition from (Π, Q_z^r) to (Π_n, P_n^r) is based on the asymptotics of the probability Q_z^r(Π_n), 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.