Universality of the limit shape of random convex lattice polygons

Bogachev, L.V. & Zarbaliev, S.M.

Let $\Pi_n$ be the set of planar convex lattice polygons $\Gamma$ (i.e., with vertices on $\mathbb{Z}_+^2$ and non-negative inclination of all edges) with fixed endpoints $0=(0,0)$ and $n=(n_1,n_2)$. We are concerned with the limit shape of a typical polygon $\Gamma\in\Pi_n$ as $n\to\infty$ with respect to a certain parametric family of probability measures $\{P_n^r\}$ ($0<r<\infty$) on the space $\Pi_n$, including the uniform distribution ($r=1$). We show that if $0<C_1\le n_2/n_1\le C_2<\infty$ then, under the scaling $(1/n_1,1/n_2)$, the limit shape is universal in the class $\{P_n^r\}$ and thus coincides with that for the uniform distribution $P^1_n$ (found independently by Vershik, Bárány, and Sinai). Our result gives a partial affirmative answer to Vershik-Prokhorov's universality conjecture. The measure $P^r_n$ is constructed, using Sinai's approach, as a conditional distribution induced by a suitable product measure $Q^r$ defined on the space $\Pi=\cup_n\Pi_n$ of polygons with a free right end. The proof involves subtle analytical tools including the Möbius inversion formula and properties of zeroes of the Riemann zeta function.

Key words:
Convex lattice polygons; limit shape; local limit theorem