Central Limit Theorem for Crossings in Randomly Embedded Graphs

Santiago Arenas-Velilla (), Octavio Arizmendi () and J. E. Paguyo ()
Additional contact information
Santiago Arenas-Velilla: University of Göttingen
Octavio Arizmendi: Centro de Investigación en Matemáticas
J. E. Paguyo: McMaster University

Journal of Theoretical Probability, 2025, vol. 38, issue 3, 1-31

Abstract: Abstract We consider the number of crossings in a random embedding of a graph, G, with vertices in convex position. We give explicit formulas for the mean and variance of the number of crossings as a function of various subgraph counts of G. Using Stein’s method and size-biased coupling, we establish an upper bound on the Kolmogorov distance between the distribution of the number of crossings and a standard normal random variable. We also consider the case where G is a random graph and obtain a Kolmogorov bound between the distribution of crossings and a Gaussian mixture distribution. As applications, we obtain central limit theorems with convergence rates for the number of crossings in random embeddings of matchings, path graphs, cycle graphs, disjoint union of triangles, random d-regular graphs, and mixtures of random graphs.

Keywords: Random graphs; Crossings; size-biased coupling; Stein’s method; Central limit theorem; Convergence rates; 60C05; 60F05 (search for similar items in EconPapers)
Date: 2025
References: Add references at CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s10959-025-01426-9 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:jotpro:v:38:y:2025:i:3:d:10.1007_s10959-025-01426-9

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/10959

DOI: 10.1007/s10959-025-01426-9

Access Statistics for this article

Journal of Theoretical Probability is currently edited by Andrea Monica

More articles in Journal of Theoretical Probability from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().