Fluctuations of lattice zonotopes and polygons

2023/03/31 by Théophile Buffière, Buffière, Théophile, Philippe Marchal +1
Mathematics · #60D05 #FOS: Mathematics #Mathematical Dynamics and Fractals #Point processes and geometric inequalities #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2303.18045

openalex publication_date 2023/03/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

Following Barany et al., who proved that large random lattice zonotopes converge to a deterministic shape in any dimension after rescaling, we establish a central limit theorem for finite-dimensional marginals of the boundary of the zonotope. In dimension 2, for large random convex lattice polygons contained in a square, we prove a Donsker-type theorem for the boundary fluctuations, which involves a two-dimensional Brownian bridge and a drift term that we identify as a random cubic curve.

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