Near-minimal spanning trees : a scaling exponent in probability models
Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 5, pp. 962-976.

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(δ 2 ). Nous montrons ce résultat d'échelle pour le modèle de la grille avec des longueurs d'arêtes aléatoires et pour le modèle euclidien.

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(δ 2 ). We prove this scaling result in the model of the lattice with random edge-lengths and in the euclidean model.

DOI : 10.1214/07-AIHP138
Classification : 05C80, 60K35, 68W40
Mots-clés : combinatorial optimization, continuum percolation, disordered lattice, local weak convergence, minimal spanning tree, Poisson point process, probabilistic analysis of algorithms, random geometric graph 
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     title = {Near-minimal spanning trees : a scaling exponent in probability models},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {962--976},
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Aldous, David J.; Bordenave, Charles; Lelarge, Marc. Near-minimal spanning trees : a scaling exponent in probability models. Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 5, pp. 962-976. doi : 10.1214/07-AIHP138. http://www.numdam.org/articles/10.1214/07-AIHP138/

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