Random Assignment Problems on 2d Manifolds

  title={Random Assignment Problems on 2d Manifolds},
  author={Dario Benedetto and Emanuele Caglioti and Sergio Caracciolo and Matteo D’Achille and Gabriele Sicuro and Andrea Sportiello},
  journal={arXiv: Mathematical Physics},
We consider the assignment problem between two sets of $N$ random points on a smooth, two-dimensional manifold $\Omega$ of unit area. It is known that the average cost scales as $E_{\Omega}(N)\sim \frac{1}{2\pi}\ln N$ with a correction that is at most of order $\sqrt{\ln N\ln\ln N}$. In this paper, we show that, within the linearization approximation of the field-theoretical formulation of the problem, the first $\Omega$-dependent correction is on the constant term, and can be exactly computed… 
1 Citations
On the quadratic random matching problem in two-dimensional domains
We investigate the average minimum cost of a bipartite matching, with respect to the squared Euclidean distance, between two samples of n i.i.d. random points on a bounded Lipschitz domain in the


On the optimal map in the 2-dimensional random matching problem
We show that, on a $2$-dimensional compact manifold, the optimal transport map in the semi-discrete random matching problem is well-approximated in the $L^2$-norm by identity plus the gradient of the
A Negative Mass Theorem for the 2-Torus
Let M be a closed surface. For a metric g on M, denote the area element by dA and the Laplace-Beltrami operator by Δ = Δg. We define the Robin mass m(p) at the point$${{p \in\, M}}$$ to be the value
Finer estimates on the $2$-dimensional matching problem
We study the asymptotic behaviour of the expected cost of the random matching problem on a $2$-dimensional compact manifold, improving in several aspects the results of L. Ambrosio, F. Stra and D.
A Negative Mass Theorem for Surfaces of Positive Genus
Let M be a closed surface. For a metric g on M, denote the Laplace-Beltrami operator by Δ = Δg. We define trace $${\Delta^{-1} = \int_M m(p)dA}$$ , where dA is the area element for g and m(p) is the
Extremals for Logarithmic Hardy-Littlewood-Sobelov Inequalities on Compact Manifolds
Abstract.Let M be a closed, connected surface and let Γ be a conformal class of metrics on M with each metric normalized to have area V. For a metric g $$\in$$ Γ, denote the area element by dV and
Minimal Discrete Energy on the Sphere
We investigate the energy of arrangements of N points on the surface of a sphere in R3, interacting through a power law potential V = rα, −2 < α < 2, where r is Euclidean distance. For α = 0, we take
A proof of Parisi’s conjecture on the random assignment problem
Abstract.An assignment problem is the optimization problem of finding, in an m by n matrix of nonnegative real numbers, k entries, no two in the same row or column, such that their sum is minimal.
Gravitational allocation on the sphere
A matching between two collections of n independent and uniform points on the sphere is defined and it is proved that the expected distance between a pair of matched points is O(log⁡n), which is optimal by a result of Ajtai, Komlós, and Tusnády.
The ζ(2) limit in the random assignment problem
  • D. Aldous
  • Mathematics
    Random Struct. Algorithms
  • 2001
The random assignment (or bipartite matching) problem asks about An=minπ ∑  i=1nc(i, π(i)), where (c(i, j)) is a n×n matrix with i.i.d. entries, say with exponential(1) distribution, and the minimum
One-loop diagrams in the Random Euclidean Matching Problem
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