Abstract

Consider the standard Gaussian measure μ on R2. Consider independent r.v.s (Xi)i≤N distributed according to μ, and an independent copy (Yi)i≤N of these r.v.s. We prove that, for some number C and N large, we have(1)(log⁡N)2C≤Einfπ⁡∑i≤Nd(Xi,Yπ(i))2≤C(log⁡N)2, where the infimum is over all permutations π of {1,…,N}. The striking point of this result is the factor (log⁡N)2. Indeed, if instead of μ we consider the uniform distribution on the unit square, it is well known that the proper factor is log⁡N. The upper bound was proved by Michel Ledoux (2017) [3].

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