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 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. We present a proof of this conjecture using representation theory of the symmetric group.

Published at http://dx.doi.org/10.1214/009117906000000223 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (4)

• Level-Spacing Distributions and the Airy Kernel
• On Orthogonal and Symplectic Matrix Ensembles
• Universality at the edge of the spectrum in Wigner random matrices
• Distribution functions for largest eigenvalues and their applications

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