Anomalous scaling of the optimal cost in the one-dimensional random assignment problem

TL;DR

This paper investigates the asymptotic behavior of the optimal matching cost in a one-dimensional random assignment problem with power-dependent costs, providing integral formulas, regularization methods, and comparisons with simulations.

Contribution

It generalizes previous results for the Euclidean assignment problem to arbitrary power costs and addresses divergence issues through a novel regularization approach.

Findings

Derived integral expression for average optimal cost for large N

Developed regularization method for divergent cases

Validated predictions with numerical simulations

Abstract

We consider the random Euclidean assignment problem on the line between two sets of $N$ random points, independently generated with the same probability density function $\varrho$. The cost of the matching is supposed to be dependent on a power $p>1$ of the Euclidean distance of the matched pairs. We discuss an integral expression for the average optimal cost for $N\gg 1$ that generalizes a previous result obtained for $p=2$. We also study the possible divergence of the given expression due to the vanishing of the probability density function. The provided regularization recipe allows us to recover the proper scaling law for the cost in the divergent cases, and possibly some of the involved coefficients. The possibility that the support of $\varrho$ is a disconnected interval is also analysed. We exemplify the proposed procedure and we compare our predictions with the results of numerical simulations.