Universality of the limit shape of convex lattice polygonal lines

Abstract

Let ${\varPi}_n$ be the set of convex polygonal lines $\varGamma$ with vertices on $\mathbb {Z}_+^2$ and fixed endpoints $0=(0,0)$ and $n=(n_1,n_2)$. We are concerned with the limit shape, as $n\to\infty$, of "typical" $\varGamma\in {\varPi}_n$ with respect to a parametric family of probability measures $\{P_n^r,0<r<\infty\}$ on ${\varPi}_n$, 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 $P^r_n$ ($r\ne1$) and $P^1_n$ are asymptotically singular. Measures $P^r_n$ are constructed, following Sinai's approach, as conditional distributions $Q_z^r(\cdot |{\varPi}_n)$, where $Q_z^r$ are suitable product measures on the space ${\varPi}=\bigcup_n{\varPi}_n$, depending on an auxiliary "free" parameter $z=(z_1,z_2)$. The transition from $({\varPi},Q_z^r)$ to $({\varPi}_n,P_n^r)$ is based on the asymptotics of the probability $Q_z^r({\varPi}_n)$, 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.