The random fractional matching problem

We consider two formulations of the random-link fractional matching problem, a relaxed version of the more standard random-link (integer) matching problem. In one formulation, we allow each node to be linked to itself in the optimal matching configuration. In the other one, on the contrary, such a link is forbidden. Both problems have the same asymptotic average optimal cost of the random-link matching problem on the complete graph. Using a replica approach and previous results of Wastlund (2010 Acta Mathematica 204 91-150), we analytically derive the finite-size corrections to the asymptotic optimal cost. We compare our results with numerical simulations and we discuss the main differences between random-link fractional matching problems and the random-link matching problem.