# Statistical Properties of the Euclidean Random Assignment Problem

Abstract : Given $2n$ points, $n$ red'' and $n$ blue'', in a Euclidean space, solving the associated Euclidean Assignment Problem consists in finding the bijection between red and blue points that minimizes a functional of the point positions. In the stochastic version of this problem, the points are a Poisson Point Process, and some interest has developed over the years on the typical and average properties of the solution in the large $n$ limit. This PhD thesis investigates this problem in a number of cases (many exact results in $d=1$, the derivation of some fine properties in $d=2$, in part still conjectural, an investigation on self-similar fractals with intermediate dimensions,...). In particular, we present new contributions on the asymptotic behavior of the cost of the solution, motivated (among others) by the physics of Disordered Systems, and by results in Functional Analysis on the Monge--Kantorovich problem in the continuum.
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Submitted on : Tuesday, January 5, 2021 - 6:55:07 PM
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• HAL Id : tel-03098672, version 1

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Matteo d'Achille. Statistical Properties of the Euclidean Random Assignment Problem. Physics [physics]. Université Paris-Saclay, 2020. English. ⟨NNT : 2020UPASQ003⟩. ⟨tel-03098672v1⟩

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