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Poisson-Dirichlet distribution for random Belyi surfaces. (English) Zbl 1113.60095

An approach to study 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 was introduced by Brooks and Makover. By Belyi’s theorem, these are “dense” in the space of compact Riemann surfaces. The question as to how these surfaces are distributed in the Teichmüller 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. The author gives a proof of this conjecture using representation theory of the symmetric group.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
05C80 Random graphs (graph-theoretic aspects)
58C40 Spectral theory; eigenvalue problems on manifolds
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