The Annals of Probability
- Ann. Probab.
- Volume 34, Number 5 (2006), 1827-1848.
Poisson–Dirichlet distribution for random Belyi surfaces
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.
Ann. Probab., Volume 34, Number 5 (2006), 1827-1848.
First available in Project Euclid: 14 November 2006
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 05C80: Random graphs [See also 60B20] 58C40: Spectral theory; eigenvalue problems [See also 47J10, 58E07]
Gamburd, Alex. Poisson–Dirichlet distribution for random Belyi surfaces. Ann. Probab. 34 (2006), no. 5, 1827--1848. doi:10.1214/009117906000000223. https://projecteuclid.org/euclid.aop/1163517226