  Brooks and Makover introduced an approach to studying the global geometric
quantities (in particular, the first eigenvalue of the Laplacian, injectivity
radius and diameter) of a ``typical'' compact Riemann surface of large genus
based on compactifying finite-area Riemann surfaces associated with random
cubic graphs; by a theorem of Belyi, these are ``dense'' in the space of
compact Riemann surfaces. The question as to how these surfaces are distributed
in the Teichm\"{u}ller spaces depends on the study of oriented cycles in random
cubic graphs with random orientation; Brooks and Makover conjectured that
asymptotically normalized cycle lengths follow Poisson--Dirichlet distribution.
We present a proof of this conjecture using representation theory of the
symmetric group.
