On the fluctuations of simple matching

For n points X"1, ..., X"n that are independent uniformly distributed over [0, 1]^2 we denote by M"n = M"n("1,...,X"n) the length of a shortest matching of these points. (That is, the points are grouped two by two, and we add the distances of each pair of matched points). We prove that for some universal constant K, we have P(|M"n - EM"n|>=K(logn)^3^2) =< n^-^2.