Euclidean Random Matching in 2D for Non-constant Densities
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Euclidean Random Matching in 2D for Non-constant Densities Dario Benedetto1
· Emanuele Caglioti1
Received: 18 December 2019 / Accepted: 2 July 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We consider the two-dimensional random matching problem in R2 . In a challenging paper, Caracciolo et al. Phys Rev E 90(1):012118 (2014), on the basis of a subtle linearization of the Monge-Ampère equation, conjectured that the expected value of the square of the Wasserstein distance, with exponent 2, between two samples of N uniformly distributed points in the unit square is log N /2π N plus corrections, while the expected value of the square of the Wasserstein distance between one sample of N uniformly distributed points and the uniform measure on the square is log N /4π N . These conjectures have been proved by Ambrosio et al. Probab Theory Rel Fields 173(1–2):433–477 (2019). Here we consider the case in which the points are sampled from a non-uniform density. For first we give formal arguments leading to the conjecture that if the density is regular and positive in a regular, bounded and connected domain in the plane, then the leading term of the expected values of the Wasserstein distances are exactly the same as in the case of uniform density, but for the multiplicative factor equal to the measure of . We do not prove these results but, in the case in which the domain is a square, we prove estimates from above that coincides with the conjectured result. Keywords Euclidean matching · Optimal transport · Monge-Ampère equation · Empirical measures Mathematics Subject Classification 60D05 · 82B44
1 Introduction Let μ be a probability distribution defined on the unit square Q = [0, 1]2 . Let us consider N and y N = { y } N of N points independently sampled from the two sets x N = {x i }i=1 i i=1
Communicated by Eric A. Carlen.
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Emanuele Caglioti [email protected] Dario Benedetto [email protected]
1
Dipartimento di Matematica, SAPIENZA Università di Roma, Piazzale Aldo Moro 5, 00185 Rome, Italy
123
D. Benedetto, E. Caglioti
distribution μ. The Euclidean Matching problem with exponent 2 consists in finding the matching i → πi , i.e. the permutation π of {1, . . . N } which minimizes the sum of the squares of the distances between x i and yπi , that is C N (x N , y N ) = min π
N
|x i − yπi |2 .
(1.1)
i=1
The cost defined above can be seen, but for a constant factor N , as the square of the 2Wasserstein distance between two probability measures. In fact, the p−Wasserstein distance W p (ν1 , ν2 ), with exponent p ≥ 1, between two probability measures ν1 and ν2 , is defined by p W p (ν1 , ν2 ) = inf Jν1 ,ν2 (dx, d y)|x − y| p , Jν1 ,ν2
where the infimum is taken on all the joint probability distributions Jν1 ,ν1 (dx, d y) with marginals with respect to dx and d y given by ν1 (dx) and ν2 (d y), respectively. Defining the empirical measures X N (dx) =
N N 1 1 δ x i (x) dx, Y N (dx) = δ yi (x) dx, N N i=1
i=1
it is possible to show that C N (x N , y N )
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