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A PDE approach to a 2-dimensional matching problem

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Abstract

We prove asymptotic results for 2-dimensional random matching problems. In particular, we obtain the leading term in the asymptotic expansion of the expected quadratic transportation cost for empirical measures of two samples of independent uniform random variables in the square. Our technique is based on a rigorous formulation of the challenging PDE ansatz by Caracciolo et al. (Phys Rev E 90:012118, 2014) that linearizes the Monge–Ampère equation.

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Acknowledgements

The first author warmly thanks S. Caracciolo for pointing out to him the paper [18] and for several conversations on the subject. He also thanks M. Ledoux for precious informations on the 1-dimensional matching problem and on the usefulness, also in this problem, of the phenomenon of concentration of measure.

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Authors and Affiliations

  1. Scuola Normale Superiore, Pisa, Italy

    Luigi Ambrosio & Federico Stra

  2. Università di Pisa, Pisa, Italy

    Dario Trevisan

Authors
  1. Luigi Ambrosio
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  2. Federico Stra
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  3. Dario Trevisan
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Corresponding author

Correspondence to Luigi Ambrosio.

Additional information

Partially supported by Project PRA_2016_41. Member of GNAMPA group (INdAM).

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Ambrosio, L., Stra, F. & Trevisan, D. A PDE approach to a 2-dimensional matching problem. Probab. Theory Relat. Fields 173, 433–477 (2019). https://doi.org/10.1007/s00440-018-0837-x

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  • DOI: https://doi.org/10.1007/s00440-018-0837-x

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