The following paper may be what you're looking for:
Schramm, Oded. Compositions of random transpositions. Israel J. Math. 147 (2005), 221--243. MR2166362 (2006h:60024)
This paper is concerned with the distribution of a random permutation in $S_n$ generated by $c n$ random transpositions (where $c>1/2$). Schramm calculates the limiting distribution of the cycle lengths (ordered from largest to smallest).
A recent paper of Nathanael Berestycki, Oded Schramm, and Ofer Zeitouni has extended the techniques of Schramm's earlier paper to the case where the random permutation is generated by random $k$-cylces. I don't know if this recent paper answers the same questions, but it might be relevant.
http://arxiv.org/abs/1001.1894https://arxiv.org/abs/1001.1894