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An easy proof of the $\zeta(2)$ limit in the random assignment problem


 
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1. Title Title of document An easy proof of the $\zeta(2)$ limit in the random assignment problem
 
2. Creator Author's name, affiliation, country Johan Wästlund; Chalmers University of Technology
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) minimum, matching, graph, exponential
 
3. Subject Subject classification 60C05, 90C27, 90C35
 
4. Description 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.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s)
 
7. Date (YYYY-MM-DD) 2009-07-05
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ecp.ejpecp.org/article/view/1475
 
10. Identifier Digital Object Identifier 10.1214/ECP.v14-1475
 
11. Source Journal/conference title; vol., no. (year) Electronic Communications in Probability; Vol 14
 
12. Language English=en
 
14. Coverage Geo-spatial location, chronological period, research sample (gender, age, etc.)
 
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