An assignment problem is the optimization problem of finding, in an m by n matrix of nonnegative real numbers, k entries, no two in the same row or column, such that their sum is minimal. Such an optimization problem is called a random assignment problem if the matrix entries are random variables. We give a formula for the expected value of the optimal k-assignment in a matrix where some of the entries are zero, and all other entries are independent exponentially distributed random variables with mean 1. Thereby we prove the formula 1 + 1/4 + 1/9 + · · · + 1/k 2 conjectured by G. Parisi for the case k = m = n, and the generalized conjecture of D. Coppersmith and G. B. Sorkin for arbitrary k, m and n.

We consider the following two-player game on a graph. A token is located at a vertex, and the players take turns to move it along an edge to a vertex that has not been visited before. A player who cannot move loses. We analyse outcomes with optimal play on percolation clusters of Euclidean lattices.On Z 2 with two different percolation parameters for odd and even sites, we prove that the game has no draws provided closed sites of one parity are sufficiently rare compared with those of the other parity (thus favoring one player). We prove this also for certain d-dimensional lattices with d ≥ 3. It is an open question whether draws can occur when the two parameters are equal.On a finite ball of Z 2 , with only odd sites closed but with the external boundary consisting of even sites, we identify up to logarithmic factors a critical window for the trade-off between the size of the ball and the percolation parameter. Outside this window, one or other player has a decisive advantage.Our analysis of the game is intimately tied to the effect of boundary conditions on maximum-cardinality matchings.

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We establish the soundness of the replica symmetric ansatz introduced by M. Mézard and G. Parisi for the minimum matching problem in the pseudo-dimension d mean field model for d ≥ 1. The case d = 1 corresponds to the π 2 /6-limit for the assignment problem proved by D. Aldous in 2001.We introduce a game-theoretical framework by which we establish the analogous limit also for d > 1.