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

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