The ζ(2) Limit in the Random Assignment Problem
نویسنده
چکیده
The random assignment (or bipartite matching) problem asks about An = minπ ∑ n i=1 c(i, π(i)), where (c(i, j)) is a n × n matrix with i.i.d. entries, say with exponential(1) distribution, and the minimum is over permutations π. Mézard and Parisi (1987) used the replica method from statistical physics to argue non-rigorously that EAn → ζ(2) = π /6. Aldous (1992) identified the limit in terms of a matching problem on a limit infinite tree. Here we construct the optimal matching on the infinite tree. This yields a rigorous proof of the ζ(2) limit and of the conjectured limit distribution of edge-costs and their rank-orders in the optimal matching. It also yields the asymptotic essential uniqueness property: every almostoptimal matching coincides with the optimal matching except on a small proportion of edges.
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