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Let ${\varPi}_n$ be the set of convex polygonal lines $\varGamma$ withvertices 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 probabilitymeasures $\{P_n^r,0<r<\infty\}$ on ${\varPi}_n$, including the uniformdistribution ($r=1$) for which the limit shape was found in the early 1990sindependently by A. M. Vershik, I. B\'ar\'any 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 $P^r_n$ areconstructed, following Sinai's approach, as conditional distributions$Q_z^r(\cdot |{\varPi}_n)$, where $Q_z^r$ are suitable product measures on thespace ${\varPi}=\bigcup_n{\varPi}_n$, 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 limittheorem. The proofs involve subtle analytical tools including the M\"obiusinversion formula and properties of zeroes of the Riemann zeta function.

Universality of the limit shape of convex lattice polygonal lines