Perfect matchings in epsilon-regular graphs and the blow-up lemma

As proved in [2], every epsilon-regular graph G on vortex set V-1 boolean O R V-2, \V-1\ = \V-2\ = n, With density d > 2 epsilon and all vertex degrees not too far from d, has about as many perfect matchings as a corresponding random bipartite graph, i.e. about d(n)n! In this paper we utilize that result to prove that with probability quickly approaching one, a perfect matching drawn randomly from G is spread evenly , in the sense that for any large subsets of vertices S subset of V-1 and T C V-2, the number of edges of the matching spanned between S and T is close to \S\\T\/n (c.f. Lemma 1). As an application we give an alternative proof of the Blow-up Lemma of Koml os, Sarkozy and Szemeredi [10].