The Annals of Probability

Poisson–Dirichlet distribution for random Belyi surfaces

Alex Gamburd

Full-text: Open access


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.

Article information

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]

Poisson–Dirichlet distribution Belyi surfaces random regular graphs


Gamburd, Alex. Poisson–Dirichlet distribution for random Belyi surfaces. Ann. Probab. 34 (2006), no. 5, 1827--1848. doi:10.1214/009117906000000223.

Export citation


  • Alon, N. (1986). Eigenvalues and expanders. Combinatorica 6 83--96.
  • Apostol, T. M. (1976). Introduction to Analytic Number Theory. Springer, New York.
  • Arratia, R., Barbour, A. D. and Tavaré, S. (2003). Logarithmic Combinatorial Structrues: A Probabilistic Approach. EMS, Zürich.
  • Belyi, G. V. (1979). Galois extensions of a maximal cyclotomic field. Izv. Akad. Nauk SSSR Ser. Mat. 43 267--276.
  • Bollobás, B. (1980). A probabilistic proof of the asymptotic formula for the number of labelled regular graphs. European J. Combin. 1 311--316.
  • Bollobás, B. (1985). Random Graphs. Academic Press Inc., London.
  • Bollobás, B. (1988). The isoperimetric number of random regular graphs. European J. Combin. 9 241--244.
  • Brooks, R. (1986). The spectral geometry of a tower of coverings. J. Differential Geom. 23 97--107.
  • Brooks, R. (1988). Some remarks on volume and diameter of Riemannian manifolds. J. Differential Geom. 27 81--86.
  • Brooks, R. (1999). Platonic surfaces. Comment. Math. Helv. 74 156--170.
  • Brooks, R. (2000). Some geometric aspects of the work of Lars Ahlfors. IMCP 14 31--39.
  • Brooks, R. and Makover, E. (2001). Riemann surfaces with large first eigenvalue. J. Anal. Math. 83 243--258.
  • Brooks, R. and Makover, E. (2001). Belyi surfaces. IMCP 15 37--46.
  • Brooks, R. and Makover, E. (2004). Random construction of Riemann surfaces. J. Differential Geom. 68 121--157.
  • Buser, P. (1978). Cubic graphs and the first eigenvalue of a Riemann surface. Math. Z. 162 87--99.
  • Buser, P. (1980). On Cheeger's inequality $\lambda_1 \ge h^2/4$. Proc. Symp. Pure. Math. 36 29--77.
  • Buser, P. (1984). On the bipartition of graphs. Discrete Appl. Math. 9 105--109.
  • Buser, P. (1992). Geometry and Spectra of Compact Riemann Surfaces. Birkhäuser, Boston.
  • Diaconis, P. (1988). Group Representations in Probability and Statistics. IMS, Hayward, CA.
  • Diaconis, P. (2000). Random walks on groups: Characters and geometry. In Groups St. Andrews 2001 in Oxford 1 120--142. Cambridge Univ. Press.
  • Diaconis, P., Mayer-Wolf, E., Zeitouni, O. and Zerner, M. (2004). The Poisson--Dirichlet law is the unique invariant distribution for uniform split-merge transformations. Ann. Probab. 32 915--938.
  • Diaconis, P. and Shahshahani, M. (1981). Generating a random permutation with random transpositions. Z. Wahrsch Verw. Gebiete 57 159--179.
  • Fomin, S. V. and Lulov, N. (1997). On the number of rim hook tableux. J. Math. Sci. 87 4118--4123.
  • Friedman, J. (1991). On the second eigenvalue and random walk in random $d$-regular graphs. Combinatorica 11 331--362.
  • Friedman, J. (2006). A proof of Alon's second eigenvalue conjecture. Memoirs of the A.M.S. To appear.
  • Gamburd, A. (2002). Spectral gap for infinite index ``congruence'' subgroups of $\mathrmSL_2(\mathbbZ)$. Israel J. Math. 127 157--200.
  • Gamburd, A. (2004). Expander graphs, random matrices, and quantum chaos. In Random Walks and Geometry (V. A. Kaimanovich, ed.) 109--141. de Gruyter, Berlin.
  • Gamburd, A. and Makover, E. (2002). On the genus of a random Riemann surface. Contemp. Math. 311 133--140.
  • Huxley, M. (1986). Eigenvalues of congruence subgroups. Contemp. Math. 53 341--349.
  • James, G. and Kerber, A. (1981). The Representation Theory of the Symmetric Group. Addison--Wesley, Reading, MA.
  • Jones, G. and Singerman, S. (1996). Belyi functions, hypermaps and Galois groups. Bull. London Math. Soc. 28 561--590.
  • Kim, H. and Sarnak, P. (2003). Refined estimates towards the Ramanujan and Selberg conjectures. J. Amer. Statist. Assoc. 16 139--183.
  • Liebeck, M. E. and Shalev, A. (2004). Fuchsian groups, coverings of Riemann surfaces, subgroups growth, random quotients and random walks. J. Algebra 276 552--601.
  • Lubotzky, A. (1994). Discrete Groups, Expanding Graphs and Invariant Measures. Birkhäuser, Basel.
  • Lubotzky, A., Phillips, R. and Sarnak, P. (1988). Ramanujan graphs. Combinatorica 8 261--277.
  • Luo, W., Rudnick, Z. and Sarnak, P. (1995). On Selberg's eigenvalue conjecture. Geom. Funct. Anal. 5 387--401.
  • McKay, B. (1981). The expected eigenvalue distribution of a large regular graph. Linear Algebra Appl. 40 203--216.
  • Mulase, M. and Penkava, M. (1998). Ribbon graphs, quadratic differentials on Riemann surfaces, and algebraic curves defined over $\mathbf\barQ$. Asian J. Math. 2 875--919.
  • Novikoff, T. Asymptotic behavior of the random 3-regular bipartite graph. Preprint.
  • Okounkov, A. (2002). Symmetric functions and random partitions. In Symmetric Functions 2001: Surveys of Developments and Perspectives (S. Fomin, ed.). Kluwer, Dordrecht.
  • Pinsker, M. (1973). On the complexity of concentrator. 7th Annual Teletrafic Conference, 318/1-318/4, Stockholm.
  • Pitman, J. Combinatorial stochastic processes. Technical Report 621, Dept. Statistics, U.C. Berkeley. Available at
  • Pippenger, N. and Schleich, K. (2006). Topological characteristics of random triangulated surfaces. Random Structures Algorithms 28 247--288.
  • Randol, B. (1974). Small eigenvalues of the Laplace operator on compact Riemann surfaces. Bull. Amer. Math. Soc. 80 996--1008.
  • Regev, A. (1981). Asymptotic values for degrees associated with strips of Young diagrams. Adv. in Math. 41 115--136.
  • Sarnak, P. (2004). What is $\ldots$ an expander? Notices Amer. Math. Soc. 51 762--763.
  • Sarnak, P. and Xue, X. (1991). Bounds for multiplicities of automorphic representations. Duke Math. J. 64 207--227.
  • Selberg, A. (1965). On the estimation of Fourier coefficients of modular forms. Proc. Symp. Pure Math. 8 1--15.
  • Shepp, L. A. and Lloyd, S. P. (1966). Ordered cycle lengths in a random permutation. Trans. Amer. Math. Soc. 121 340--357.
  • Sinai, Ya. and Soshnikov, A. (1998). A refinement of Wigner's semicircle law in a neighborhood of the spectrum edge for random symmetric matrices. Funct. Anal. Appl. 32 114--131.
  • Soshnikov, A. (1999). Universality at the edge of the spectrum in Wigner random matrices. Comm. Math. Phys. 207 697--733.
  • Tsilevich, N. V. (2000). Stationary random partitions of a natural series. Theory Probab. Appl. 44 60--74.
  • Tracy, C. A. and Widom, H. (1994). Level-spacing distribution and the Airy kernel. Comm. Math. Phys. 159 151--174.
  • Tracy, C. A. and Widom, H. (1996). On orthogonal and symplectic matrix ensembles. Comm. Math. Phys. 177 727--754.
  • Tracy, C. A. and Widom, H. (2002). Distribution functions for largest eigenvalues and their applications. In Proceedings of the International Congress of Mathematicians, Beijing I (L. I. Tatsien, ed.) 587--596. Higher Education Press, Beijing.
  • Vershik, A. M. (1989). A partition function connected with Young diagrams. J. Soviet Math. 47 2379--2386.
  • Vershik, A. M. and Shmidt, A. A. (1977). Limit measures that arise in the asymptotic theory of symmetric groups. I. Teor. Verojatnost. i Primenen. 22 72--88.
  • Vershik, A. M. and Shmidt, A. A. (1978). Limit measures that arise in the asymptotic theory of symmetric groups. II. Teor. Verojatnost. i Primenen. 23 42--54.
  • Wormald, N. C. (1999). Models of random regular graphs. In Survey in Combinatorics 1999 239--298. Cambridge Univ. Press.