Poisson--Dirichlet distribution for random Belyi surfaces

Abstract

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.