COMPLETE CONVERGENCE OF THE DIRECTED TSP

Consider the random directed graph G(n) whose vertices X1,..., X(n) are independent uniformly distributed over [0, 1]2. For 1 less-than-or-equal-to i < j less-than-or-equal-to n, the orientation of the edge X(i)X(j) is selected at random, independently for each edge and independently of the X(i)'s. Denote by U(n) the length of the shortest path through G(n). Then for some constant beta > 0 and all epsilon > 0 we have SIGMA-n greater-than-or-equal-to 1P(\n-1/2U(n) - beta\ > epsilon) < infinity.