Inverse Limit Shape Problem for Multiplicative Ensembles of Convex Lattice Polygonal Lines

Abstract

Convex polygonal lines with vertices in Z+2 and endpoints at 0=(0,0) and n=(n1,n2)→∞, such that n2/n1→c∈(0,∞), under the scaling n1−1, have limit shape γ* with respect to the uniform distribution, identified as the parabola arc c(1−x1)+x2=c. This limit shape is universal in a large class of so-called multiplicative ensembles of random polygonal lines. The present paper concerns the inverse problem of the limit shape. In contrast to the aforementioned universality of γ*, we demonstrate that, for any strictly convex C3-smooth arc γ⊂R+2 started at the origin and with the slope at each point not exceeding 90∘, there is a sequence of multiplicative probability measures Pnγ on the corresponding spaces of convex polygonal lines, under which the curve γ is the limit shape.