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

Leonid V. Bogachev and Sakhavet M. Zarbaliev ()
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Leonid V. Bogachev: Department of Statistics, School of Mathematics, University of Leeds, Woodhouse Lane, Leeds LS2 9JT, UK
Sakhavet M. Zarbaliev: Department of Mathematics, Econometrics and Information Technology, School of International Economic Relations, MGIMO University, Prospekt Vernadskogo 76, Moscow 119454, Russia

Mathematics, 2023, vol. 11, issue 2, 1-23

Abstract: Convex polygonal lines with vertices in Z + 2 and endpoints at 0 = ( 0 , 0 ) and n = ( n 1 , n 2 ) → ∞ , such that n 2 / n 1 → c ∈ ( 0 , ∞ ) , under the scaling n 1 − 1 , have limit shape γ * with respect to the uniform distribution, identified as the parabola arc c ( 1 − x 1 ) + x 2 = 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 C 3 -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 P n γ on the corresponding spaces of convex polygonal lines, under which the curve γ is the limit shape.

Keywords: convex lattice polygonal lines; limit shape; multiplicative probability measure; local limit theorem (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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