Almost sure convergence of the minimum bipartite matching functional in Euclidean space

الأصل · EN

Let Lₙ = LMBM(X₁,..., Xₙ; Y₁,..., Yₙ) be the minimum length of a bipartite matching between two sets of points in Rᵈ, where X₁,..., Xₙ,... and Y₁,..., Yₙ,... are random points independently and uniformly distributed in [0,1]ᵈ. We prove that for d ≥ 3, Lₙ/N¹⁻¹/ᵈ converges with probability one to a constant βMBM(d)>0 as N→ ∞.