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

Johan Wästlund (Chalmers University of Technology)

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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Pages: 261-269

Publication Date: July 5, 2009

DOI: 10.1214/ECP.v14-1475

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