Random Matchings in Regular Graphs

d

-regular graph G, let M be chosen uniformly at random from the set of all matchings of G, and for let be the probability that M does not cover x.

We show that for large d, the 's and the mean μ and variance of are determined to within small tolerances just by d and (in the case of μ and ) :

Theorem. For any d-regular graph G,

(a) , so that ,

(b) ,

where the rates of convergence depend only on d.