Near-minimal spanning trees: A scaling exponent in probability models
David J. Aldous, Charles Bordenave, and Marc Lelarge
Source: Ann. Inst. H. Poincaré Probab. Statist. Volume 44, Number 5
(2008), 962-976.
Abstract
We study the relation between the minimal spanning tree (MST) on many random points and the “near-minimal” tree which is optimal subject to the constraint that a proportion δ of its edges must be different from those of the MST. Heuristics suggest that, regardless of details of the probability model, the ratio of lengths should scale as 1+Θ(δ2). We prove this scaling result in the model of the lattice with random edge-lengths and in the Euclidean model.
Résumé
Nous étudions la relation entre l’arbre couvrant minimal (ACM) sur des points aléatoires et l’arbre “quasi” optimal sous la contrainte qu’une proportion δ de ses arêtes soit différente de celles de l’ACM. Un raisonnement heuristique suggère que quelque soit le modèle probabiliste sous-jacent, le ratio des longueurs des deux arbres doit varier en 1+Θ(δ2). Nous montrons ce résultat d’échelle pour le modèle de la grille avec des longueurs d’arêtes aléatoires et pour le modèle Euclidien.
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aihp/1222261920
Digital Object Identifier: doi:10.1214/07-AIHP138
Zentralblatt MATH identifier: 05611469
Mathematical Reviews number (MathSciNet): MR2453778
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