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September 2006 Poisson–Dirichlet distribution for random Belyi surfaces
Alex Gamburd
Ann. Probab. 34(5): 1827-1848 (September 2006). DOI: 10.1214/009117906000000223


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.


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Alex Gamburd. "Poisson–Dirichlet distribution for random Belyi surfaces." Ann. Probab. 34 (5) 1827 - 1848, September 2006.


Published: September 2006
First available in Project Euclid: 14 November 2006

zbMATH: 1113.60095
MathSciNet: MR2271484
Digital Object Identifier: 10.1214/009117906000000223

Primary: 60K35
Secondary: 05C80 , 58C40

Keywords: Belyi surfaces , Poisson–Dirichlet distribution , Random regular graphs

Rights: Copyright © 2006 Institute of Mathematical Statistics


Vol.34 • No. 5 • September 2006
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