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

J. H. Boutet de Monvel·O. C. Martin
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Abstract

Let $L_N = L_{MBM}(X_1,..., X_N; Y_1,..., Y_N)$ be the minimum length of a bipartite matching between two sets of points in $\mathbf{R}^d$, where $X_1,..., X_N,...$ and $Y_1,..., Y_N,...$ are random points independently and uniformly distributed in $[0,1]^d$. We prove that for $d \ge 3$, $L_N/N^{1-1/d}$ converges with probability one to a constant $β_{MBM}(d)>0$ as $N\to \infty $.

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J. H. Boutet de Monvel, O. C. Martin. 2002-05-13. Almost sure convergence of the minimum bipartite matching functional in Euclidean space. https://arxiv.org/abs/math/0205140

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