Near-minimal spanning trees: A scaling exponent in probability models - Aldous, David J.; Bordenave, Charles; Lelarge, Marc (Annales de l'I.H.P. Probabilités et statistiques, 2008)
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.