One, Two And Three Times Log N/N For Paths In A Complete Graph With Random Weights

Consider the minimal weights of paths between two points in a complete graph K n with random weights on the edges, the weights being e.g. uniformly distributed. It is shown that, asymptotically, this is log n=n for two given points, that the maximum if one point is xed and the other varies is 2 log n=n, and that the maximum over all pairs of points is 3 log n=n. Some further related results are given too, including results on asymp-totic distributions and moments, and on the number of edges in the minimal weight paths. 1. Introduction Let a random weight T ij be assigned to every edge ij of the complete graph K n. (Thus T ji = T ij. We do not deene T ij for i = j.) We assume that the