The $\zeta(2)$ Limit in the Random Assignment Problem David Aldous The random assignment (or bipartite matching) problem asks about $ A_n = \min_\pi \sum_{i=1}^n c(i,\pi(i)) $, where $(c(i,j))$ is a $n \times n$ matrix with i.i.d. entries, say with exponential(1) distribution, and the minimum is over permutations $\pi$. M\'{e}zard and Parisi (1987) used the replica method from statistical physics to argue non-rigorously that $EA_n \to \zeta(2) = \pi^2/6$. 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 $\zeta(2)$ 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 almost-optimal matching coincides with the optimal matching except on a small proportion of edges.