# Finite-size corrections in the random assignment problem

## TL;DR

This paper analytically derives finite-size corrections to the average optimal cost in the random assignment problem using the replica formalism, revealing how these corrections vary with different cost distributions.

## Contribution

It provides the first analytical derivation of finite-size corrections for a broad class of cost distributions in the assignment problem, including power-law, Gamma, and delta functions.

## Key findings

- Corrections change sign and scaling when moving from power-law to Gamma distributions.
- Numerical solutions confirm the analytical predictions.
- Behavior near delta-function distribution is analyzed.

## Abstract

We analytically derive, in the context of the replica formalism, the first finite size corrections to the average optimal cost in the random assignment problem for a quite generic distribution law for the costs. We show that, when moving from a power-law distribution to a $\Gamma$ distribution, the leading correction changes both in sign and in its scaling properties. We also examine the behavior of the corrections when approaching a $\delta$-function distribution. By using a numerical solution of the saddle-point equations, we provide predictions that are confirmed by numerical simulations.