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2009 An easy proof of the $\zeta(2)$ limit in the random assignment problem
Johan Wästlund
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Electron. Commun. Probab. 14: 261-269 (2009). DOI: 10.1214/ECP.v14-1475

Abstract

The edges of the complete bipartite graph $K_{n,n}$ are given independent exponentially distributed costs. Let $C_n$ be the minimum total cost of a perfect matching. It was conjectured by M. Mézard and G. Parisi in 1985, and proved by D. Aldous in 2000, that $C_n$ converges in probability to $\pi^2/6$. We give a short proof of this fact, consisting of a proof of the exact formula $1 + 1/4 + 1/9 + \dots + 1/n^2$ for the expectation of $C_n$, and a $O(1/n)$ bound on the variance.

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Johan Wästlund. "An easy proof of the $\zeta(2)$ limit in the random assignment problem." Electron. Commun. Probab. 14 261 - 269, 2009. https://doi.org/10.1214/ECP.v14-1475

Information

Accepted: 5 July 2009; Published: 2009
First available in Project Euclid: 6 June 2016

zbMATH: 1195.60018
MathSciNet: MR2516261
Digital Object Identifier: 10.1214/ECP.v14-1475

Subjects:
Primary: 60C05
Secondary: 90C27, 90C35

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