Scaling and non-standard matching theorems

Michel Talagrand

doi:10.1016/j.crma.2018.04.018

Probability theory

Scaling and non-standard matching theorems

Presented by: Gilles Pisier

Michel Talagrand ¹

¹ 23, rue Louis-Pouey, 92800 Puteaux, France

Comptes Rendus. Mathématique, Volume 356 (2018) no. 6, pp. 692-695.

Abstract
Résumé

Consider the standard Gaussian measure μ on $R^{2}$ . Consider independent r.v.s ${(X_{i})}_{i \leq N}$ distributed according to μ, and an independent copy ${(Y_{i})}_{i \leq N}$ of these r.v.s. We prove that, for some number C and N large, we have

\frac{{(\log N)}^{2}}{C} \leq E \inf_{π} \sum_{i \leq N} d {(X_{i}, Y_{π (i)})}^{2} \leq C {(\log N)}^{2},

(1)

where the infimum is over all permutations π of

{1, \dots, 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].

Considérons une suite indépendente ${(X_{i})}_{i \leq N}$ de variables aléatoires distribuées comme la mesure gaussienne canonique μ sur $R^{2}$ et une copie independente ${(Y_{i})}_{i \leq N}$ de cette même suite. Pour une certaine constante universelle C et $N \geq 2$ , nous avons les inégalités

\frac{{(\log N)}^{2}}{C} \leq E \inf_{π} \sum_{i \leq N} d {(X_{i}, Y_{π (i)})}^{2} \leq C {(\log N)}^{2}

(1)

où l'infimum est pris sur toutes les permutations π de

{1, \dots, N}

. La borne supérieure a été prouvée par Michel Ledoux (2017) [3], qui conjecturait que l'inégalité (1) était correcte avec un facteur

\log N

et non pas

{(\log N)}^{2}

. C'est précisement l'apparence de ce facteur

{(\log N)}^{2}

qui est non standard.

Received: 2018-04-11
Accepted: 2018-04-13
Published online: 2018-05-25

DOI: 10.1016/j.crma.2018.04.018

Author's affiliations:

Michel Talagrand ¹

¹ 23, rue Louis-Pouey, 92800 Puteaux, France

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     author = {Michel Talagrand},
     title = {Scaling and non-standard matching theorems},
     journal = {Comptes Rendus. Math\'ematique},
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     number = {6},
     year = {2018},
     doi = {10.1016/j.crma.2018.04.018},
     language = {en},
}

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Michel Talagrand. Scaling and non-standard matching theorems. Comptes Rendus. Mathématique, Volume 356 (2018) no. 6, pp. 692-695. doi : 10.1016/j.crma.2018.04.018. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.04.018/

References
Cited by

[1] M. Ajtai; J. Komlós; G. Tusnády On optimal matchings, Combinatorica, Volume 4 (1984) no. 4, pp. 259-264

[2] L. Ambrosio; F. Stra; D. Trevisan A PDE approach to a 2-dimensional matching problem, Probab. Theory Relat. Fields (2016) (in press)

[3] M. Ledoux On optimal matching of Gaussian samples, Zap. Nauč. Semin. POMI, Volume 457 (2017) (Veroyatnost' i Statistika 25 226–264)

[4] M. Talagrand Upper and Lower Bounds for Stochastic Processes http://michel.talagrand.net/ULB.pdf (new edition in preparation, available at)

[5] J. Yukich Some generalizations of the Euclidean two-sample matching problem, Probability in Banach Spaces, 8, Progress in Probability, vol. 30, Birkhäuser, 1992, pp. 55-66

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