Poisson-Dirichlet distribution for random Belyi surfaces

• 成果类型: Article
• 署名作者: Gamburd, Alex
• 署名单位: University of California System; University of California Santa Cruz
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117906000000223
• 发表日期: 2006
• 页码: 1827-1848
• 关键词: riemann surfaces; Random Permutation; regular graphs; random-walks; EIGENVALUE; coverings; spectrum; matrices; edge; LAW

摘要

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 Teichmuller 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.