We present a linear expected time algorithm for finding maximum cardinality matchings in sparse random graphs. This is optimal and improves on previous results by a logarithmic factor.

An n-lift of a digraph K is a digraph with a vertex set $V(K)\times [n]$, and for each directed edge $(i,j)\in E(K)$ there is a perfect matching between fibers $\{i\}\times [n]$ and $\{j\}\times [n]$, with edges directed from fiber i to fiber j. If these matchings are chosen independently and uniformly at random, then we say that we have a random n-lift. We show that if h is sufficiently large, then a random n-lift of the complete digraph $\DK_h$ is a Hamiltonian .