Scaling and non-standard matching theorems

Consider the standard Gaussian measure mu on R-2. Consider independent r.v.s (X-i)i <= N distributed according to mu, and an independent copy (Y-i)(i <= N) of these r.v.s. We prove that, for some number C and N large, we have (log N)(2)/C <= E inf(pi)Sigma(i <= N)d(Xi,Y pi(i))(2 )<= C(log N)(2), where the infimum is over all permutations pi of {1,..., N}. The striking point of this result is the factor (logN)(2). Indeed, if instead of mu we consider the uniform distribution on the unit square, it is well known that the proper factor is logN. The upper bound was proved by Michel Ledoux (2017) [3]. (C) 2018 Academie des sciences. Published by Elsevier Masson SAS. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).