title: One-loop diagrams in the random Euclidean matching problem.
abstract: The matching problem is a notorious combinatorial optimization problem that has attracted for many years the attention of the statistical physics community. Here we analyze the Euclidean version of the problem, i.e., the optimal matching problem between points randomly distributed on a d-dimensional Euclidean space, where the cost to minimize depends on the points' pairwise distances. Using Mayer's cluster expansion we write a formal expression for the replicated action that is suitable for a saddle point computation. We give the diagrammatic rules for each term of the expansion, and we analyze in detail the one-loop diagrams. A characteristic feature of the theory, when diagrams are perturbatively computed around the mean field part of the action, is the vanishing of the mass at zero momentum. In the non-Euclidean case of uncorrelated costs instead, we predict and numerically verify an anomalous scaling for the sub-sub-leading correction to the asymptotic average cost.
authors.affiliation: Politecnico di Torino, Corso Duca degli Abruzzi, 24, I-10129 Torino, Italy and Human Genetics Foundation - Torino, Via Nizza 52, I-10126 Torino, Italy.
authors.forename: Carlo
authors.initials: C
authors.lastname: Lucibello
authors.affiliation: Dipartimento di Fisica, INFN - Sezione di Roma1, CNR-IPCF UOS Roma Kerberos, Università "Sapienza", P.le A. Moro 2, I-00185, Rome, Italy.
authors.forename: Giorgio
authors.initials: G
authors.lastname: Parisi
authors.affiliation: Centro Brasileiro de Pesquisas Físicas, Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro - RJ, Brazil.
authors.forename: Gabriele
authors.initials: G
authors.lastname: Sicuro