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


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.


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

Primary: 60C05
Secondary: 90C27 , 90C35

Keywords: exponential , graph , Matching , minimum

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