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## Poisson-Dirichlet distribution for random Belyi surfaces (2006)

Venue: | Ann. Probab |

Citations: | 8 - 0 self |

### Citations

2342 | Random Graphs.
- Bollobas
- 2001
(Show Context)
Citation Context ...result of Pinsker on expansion coefficient of random regular graphs was considerably strengthened by Bollobás [7] who also introduced a widely used configuration model model for random regular graphs =-=[6]-=-. In this model random k-regular graphs on N vertices are represented as the images of so-called configurations. Let W = ⋃n j=1 Wj be a fixed set of 2m = nd vertices, where |Wj| = d. A configuration F... |

808 |
Introduction to Analytic Number Theory
- Apostol
- 1976
(Show Context)
Citation Context ... N − λ1, we can now estimate Σ1 using (24): (25) Σ1 ≤ ∑ m≤r≤ N 4 p(r) ) t , ( N−r r where p(r) is the number of partitions of r. Since the number of partitions p(r) satisfies the following inequality =-=[2]-=- valid for all r ≥ 1 (26) p(r) ≤ exp(π √ 2r/3), we have (27) Σ1 ≤ ∑ m≤r≤ N 4 c √ r 1 ( ) N−r t r for absolute constant c1 = e π √ 2/3. Let ar = 1 ( N−r r ) ar+1 = (N − r)(r + 1) (N − 2r)(N − 2r − 1) a... |

621 |
The representation theory of the symmetric groups,
- James
- 1978
(Show Context)
Citation Context ...3) ‖Pk ∗ P2 − U‖ 2 ≤ 1 4 ∑ ρ∈ ÂN ρ̸=id ( χ ρ (Ck)χ ρ (C2) dim(ρ) ) 2 . The representation theory of the alternating group AN is closely allied with the representation theory of the symmetric group SN =-=[28]-=-. Representations of the symmetric group SN are labelled by partitions λ ⊢ N. A partition λ of a nonnegative integer N is a sequence (λ1, . . .,λr) ∈ Nr satisfying λ1 ≥ · · · ≥ λr and ∑ λi = N. We cal... |

422 | Level-spacing distribution and the Airy kernel,
- Tracy
- 1994
(Show Context)
Citation Context ...gest eigenvalue of the adjacency matrix of random regular graph, suitably rescaled, follows Tracy-Widom GOE distribution. xPOISSON-DIRICHLET DISTRIBUTION FOR RANDOM BELYI SURFACES 21 Tracy and Widom =-=[49, 50]-=- computed the limiting distribution function for the largest eigenvalue in the classical Gaussian ensembles; these distributions functions are expressed in terms of certain Painlevé II function and ar... |

399 | Eigenvalues and expanders,
- Alon
- 1996
(Show Context)
Citation Context ...alized cycles lengths follow Poisson-Dirichlet distribution. We recall the definition of Poisson-Dirichlet distribution [3]. Let B1, B2, . . . be independent random variables uniformly distributed on =-=[0, 1]-=-. Define G = (G1, G2, . . ...) as follows: G1 = B1; G2 = (1−B1)B2; . . .Gi = (1−B1)(1−B2) . . .(1−Bi−1)Bi. The random sequence G can be viewed as a description of a random breaking of a stick of unit ... |

399 | and P.Sarnak. Ramanujan graphs.
- Lubotzky, Phillips
- 1988
(Show Context)
Citation Context ...es Gn,k are asymptotically Ramanujan: for k fixed and ε > 0, the probability that λ1(Xn,k) ≤ 2 √ k − 1 + ε tends to 1 as n → ∞. The bound of 2 √ k − 1 is optimal in view of the result of Alon–Boppana =-=[1, 35]-=-. We also mention an early result of McKay [37], who showed that spectral density of random k-regular graphs converges to Kesten’s measure, that is, a measure supported on [−2 √ k − 1,2 √ k − 1] and g... |

325 |
Geometry and Spectra of Compact Riemann Surfaces,
- Buser
- 1992
(Show Context)
Citation Context ...interactions. 1. Introduction Study of the first eigenvalue of the Laplace operator on compact Riemann surfaces of increasing genus has received considerable attention over the last thirty years; see =-=[18]-=- and references therein. On the one hand, we have a celebrated theorem of Selberg [46] (see [34] and [30] for refined estimates towards Selberg’s conjecture) and its generalization by Sarnak and Xue [... |

284 |
A probabilistic proof of an asymptotic formula for the number of labelled regular graphs
- Bollobás
- 1980
(Show Context)
Citation Context ...m) = 2 2 = ) . . . N(m) (2m − 1)(2m − 3) . . .(2m − 2l + 1) ( ) 2 /(m − l)! 2 Using configuration model and in particular (6) Bollobás proved the following result.10 ALEX GAMBURD Theorem 2 (Bollobás =-=[5]-=-). Let Xi denote the number of closed walks in G ∈ Gn,k of length i. Then for i = o(log k−1 n) as n → ∞ random variables Xi converge to independent Poisson random variables with mean (k−1)i 2i . Count... |

258 | On orthogonal and symplectic matrix ensembles.
- Tracy, Widom
- 1996
(Show Context)
Citation Context ...gest eigenvalue of the adjacency matrix of random regular graph, suitably rescaled, follows Tracy-Widom GOE distribution. xPOISSON-DIRICHLET DISTRIBUTION FOR RANDOM BELYI SURFACES 21 Tracy and Widom =-=[49, 50]-=- computed the limiting distribution function for the largest eigenvalue in the classical Gaussian ensembles; these distributions functions are expressed in terms of certain Painlevé II function and ar... |

219 | Combinatorial Stochastic Processes.
- Pitman
- 2006
(Show Context)
Citation Context ...nsity θ(1 − x) θ−1 on [0,1] with θ > 0, the resulting distribution is called Poisson–Dirichlet distribution with parameter θ. Poisson–Dirichlet distribution arises in a great variety of problems; see =-=[3, 42]-=- and references therein. In i=1POISSON–DIRICHLET DISTRIBUTION FOR RANDOM BELYI SURFACES 3 a recent breakthrough work [21], Diaconis, Mayer-Wolf, Zeitouni and Zerner proved a conjecture of Vershik [52... |

214 | Models of random regular graphs
- Wormald
- 1999
(Show Context)
Citation Context ...er three faces are unchanged, hence F = 4, and the surface has genus 1. 3. Random regular graphs. In this section we briefly review the pertinent facts on random k-regular graphs; see Wormald’s paper =-=[59]-=- for an excellent survey. Given a k-regular graph G and a subset X of V , the expansion of X, c(X), is defined to be the ratio |∂(X)|/|X|, where ∂(X) = {y ∈ G :distance(y,X) = 1}. The expansion coeffi... |

180 |
Generating a random permutation with random transpositions.
- Diaconis, Shahshahani
- 1981
(Show Context)
Citation Context ... and α is chosen with uniform probability on the conjugacy class consisting of the product of 2-cycles in the symmetric group SN with N = nk. In section 4 using Diaconis-Shahshahani upper-bound lemma =-=[21]-=-, the estimate on the number of rim hook tableux by Fomin and Lulov [22], and representation theory of the symmetric group (in particular, hook-length formula and Murnaghan-Nakayama rule), we show tha... |

165 | A proof of Alon’s second eigenvalue conjecture and related problems
- Friedman
(Show Context)
Citation Context ... random variables Xi converge to independent Poisson random variables with mean (k−1)i 2i . Counting cycles of length greater than log n is substantially more difficult. In a recent breakthrough work =-=[24]-=- Friedman estimates the number of cycles of length O(log 2 n) and uses this estimates (among other things) to prove that k-regular graphs on n vertices Gn,k are asymptotically Ramanujan: for k fixed a... |

157 |
On galois extensions of a maximal cyclotomic field
- Belyi
- 1980
(Show Context)
Citation Context ...rmal compactification of S O (Γ, O); Brooks and Makover proved that almost always the global geometry of S C (Γ, O) is controlled by the geometry of S O (Γ, O). Moreover, according to Belyi’s theorem =-=[4]-=- the surfaces S C (Γ, O) are precisely the Riemann surfaces which can be defined over some number field and so form a “dense” set in the space of all Riemann surfaces. The question as to how these sur... |

150 | Universality at the edge of the spectrum in Wigner random matrices.
- Soshnikov
- 1999
(Show Context)
Citation Context ....52 as the size of the graph tends to infinity, corresponding to the skewness in the Tracy-Widom GOE distribution. To approach Conjecture 1 following the method Sinai and Soshnikov [47] and Soshnikov =-=[48]-=- in his breakthrough proof of the universality at the edge of the spectrum in Wigner matrices, one needs precise information for the number of closed walks of size up to n 2/3 , where n is the size of... |

146 |
On the estimation of Fourier coefficients of modular forms
- Selberg
(Show Context)
Citation Context ... compact Riemann surfaces of increasing genus has received considerable attention over the last thirty years; see [18] and references therein. On the one hand, we have a celebrated theorem of Selberg =-=[46]-=- (see [34] and [30] for refined estimates towards Selberg’s conjecture) and its generalization by Sarnak and Xue [45] (see [27] and [25] for related results), asserting that the first eigenvalue of th... |

114 |
Asymptotic values for degrees associated with strips of Young tableau,
- Regev
- 1981
(Show Context)
Citation Context ... λ⊢N λ1,λ ′ 1 ≤N−m (f λ ) −t = O(N −mt ), where implied constant depends only on m. We remark that the sums of the form ∑ λ⊢N (fλ) β for β > 0 have been studied by Vershik and Kerov [52] and by Regev =-=[43]-=-. In particular, Regev relates the asymptotic computations of such sums to the matrix integral of the form ∫ ∞ ∫ ∞ ∏ . . . −∞ |xi − xj| −∞ i<j β e −(β/2)(x2 1 +···+x2 N ) dx1 . . .dxN, this being perh... |

105 |
The expected eigenvalue distribution of a large regular graph
- McKay
- 1981
(Show Context)
Citation Context ...d and ε > 0 the probability that λ1(Xn,k) ≤ 2 √ k − 1 + ε tends to 1 as n → ∞. The bound of 2 √ k − 1 is optimal in view of the result of AlonBoppana [1, 33]. We also mention an early result of McKay =-=[35]-=- who showed that spectral density of random k-regular graphs converges to Kesten’s measure, that is a measure supported on [−2 √ k − 1, 2 √ k − 1] and given by (7) νk = k √ 4(k − 1) − t2 2π k2 − t2 . ... |

79 | On the second eigenvalue and random walks in random d-regular graphs,
- Friedman
- 1991
(Show Context)
Citation Context ...dependent Poisson random variables with mean (k−1)i 2i . Counting cycles of length greater than logn is substantially more difficult. In a recent breakthrough work [25], following his earlier work in =-=[24]-=-, Friedman estimates the number of cycles of length O(log 2 n) and uses this estimate (among other things) to prove that k-regular graphs on n vertices Gn,k are asymptotically Ramanujan: for k fixed a... |

74 |
Ordered cycle lengths in a random permutation
- Shepp, Lloyd
- 1966
(Show Context)
Citation Context ...istribution. We then invoke what is perhaps the oldest occurrence of Poisson–Dirichlet distribution— the distribution of normalized cycle lengths for a random permutation in Sn as n tends to infinity =-=[49, 57, 58]-=-—to prove the conjecture of Brooks and Makover. It turns out that the number of oriented cycles in random cubic graphs with random orientation was also studied by Pippenger and Schleich [43] in connec... |

73 | On the complexity of a concentrator
- Pinsker
- 1973
(Show Context)
Citation Context ...screte Cheeger-Buser inequality, the condition (3) can be rewritten in terms of of the second largest eigenvalue of the adjacency matrix A(G) as follows: (4) lim sup λ1(An,k) < k. n→∞ In 1973 Pinsker =-=[40]-=- observed that a random regular graph is a good expander. This corresponds to the following fact about random matrices: random symmetric matrix of size N with k ones in each row and column and all oth... |

72 |
Models of random regular graphs. Surveys in Combinatorics
- Wormald
- 1999
(Show Context)
Citation Context ...ree faces are unchanged hence, F = 4, and the surface will have genus 1 . 3. Random regular graphs In this section we briefly review the pertinent facts on random kregular graphs; see Wormald’s paper =-=[53]-=- for an excellent survey. Given a k-regular graph G and a subset X of V , the expansion of X, c(X), is defined to be the ratio |∂(X)|/|X|, where ∂(X) = {y ∈ G : distance(y, X) = 1}. The expansion coef... |

69 |
Discrete Groups, Expanding Graphs
- Lubotzky
- 1994
(Show Context)
Citation Context ...aph G is an analogue of the Cheeger’s constant for Riemann surfaces and is defined as follows { (2) c(G) = inf c(X) | |X| < 1 2 |G| } . A family of k-regular graphs Xn,k forms a family of C-expanders =-=[32, 44]-=- if there is a fixed positive constant C, such that (3) lim inf n→∞ c(Xn,k) ≥ C.POISSON-DIRICHLET DISTRIBUTION FOR RANDOM BELYI SURFACES 9 The adjacency matrix of G, A(G) is the |G | by |G | matrix, ... |

67 |
The isoperimetric number of random regular graphs,
- Bollobas
- 1988
(Show Context)
Citation Context ...t the next eigenvalue will be bounded away from k by a fixed amount independent of N. The result of Pinsker on expansion coefficient of random regular graphs was considerably strengthened by Bollobás =-=[7]-=- who also introduced a widely used configuration model model for random regular graphs [6]. In this model random k-regular graphs on N vertices are represented as the images of so-called configuration... |

61 |
Refined estimates towards the Ramanujan and Selberg Conjectures.
- Kim, Sarnak
- 2003
(Show Context)
Citation Context ...rfaces of increasing genus has received considerable attention over the last thirty years; see [18] and references therein. On the one hand, we have a celebrated theorem of Selberg [46] (see [34] and =-=[30]-=- for refined estimates towards Selberg’s conjecture) and its generalization by Sarnak and Xue [45] (see [27] and [25] for related results), asserting that the first eigenvalue of the congruence surfac... |

56 |
Bounds for multiplicities of automorphic representations
- Sarnak, Xue
- 1991
(Show Context)
Citation Context ...] and references therein. On the one hand, we have a celebrated theorem of Selberg [46] (see [34] and [30] for refined estimates towards Selberg’s conjecture) and its generalization by Sarnak and Xue =-=[45]-=- (see [27] and [25] for related results), asserting that the first eigenvalue of the congruence surfaces of arbitrary genus is bounded away from zero; on the other hand there are examples due to Selbe... |

51 |
A refinement of Wigner's semicircle law in a neighborhood of the spectrum edge for random symmetric matrices, Funct
- Sinai, Soshnikov
- 1998
(Show Context)
Citation Context ...anujan approaches 0.52 as the size of the graph tends to infinity, corresponding to the skewness in the Tracy-Widom GOE distribution. To approach Conjecture 1 following the method Sinai and Soshnikov =-=[47]-=- and Soshnikov [48] in his breakthrough proof of the universality at the edge of the spectrum in Wigner matrices, one needs precise information for the number of closed walks of size up to n 2/3 , whe... |

49 | Ribbon graphs, quadratic differentials on Riemann surfaces, and algebraic curves defined over Q̄ ,” [math-ph/9811024
- Mulase, Penkava
(Show Context)
Citation Context ... S → C unramified outside {0, 1, ∞}. We call such surfaces Belyi surfaces. In this section we review BrooksMakover construction of Belyi surfaces from cubic graphs. We remark that Mulase and Penakava =-=[37]-=- have given an alternative very interesting construction of Belyi surfaces; in their construction the edges of the graphs are allowed to have variable lengths.POISSON-DIRICHLET DISTRIBUTION FOR RANDO... |

44 | Belyi functions, hypermaps and Galois groups
- Jones, Singerman
- 1996
(Show Context)
Citation Context ...is a Belyi surface if and only if one can find finitely many points {p1, . . .,pl} on S such that S − {p1, . . .,pl} is isomorphic to H 2 /G where G is a torsion-free finite index subgroup of PSL2(Z) =-=[29]-=-, the lemma is proved. We define probability on the space of oriented graphs with n-vertices (Γn, O) as follow: We pick a random cubic graph with n vertices using the Bollobas model, described in the ... |

42 |
On the spectral gap for infinite index congruence subgroups of SL2(Z
- Gamburd
(Show Context)
Citation Context ...erein. On the one hand, we have a celebrated theorem of Selberg [46] (see [34] and [30] for refined estimates towards Selberg’s conjecture) and its generalization by Sarnak and Xue [45] (see [27] and =-=[25]-=- for related results), asserting that the first eigenvalue of the congruence surfaces of arbitrary genus is bounded away from zero; on the other hand there are examples due to Selberg [46], Randol [42... |

41 |
On Selberg’s eigenvalue conjecture.
- Luo, Rudnick, et al.
- 1995
(Show Context)
Citation Context ...iemann surfaces of increasing genus has received considerable attention over the last thirty years; see [18] and references therein. On the one hand, we have a celebrated theorem of Selberg [48] (see =-=[36]-=- and [32] for refined estimates toward Selberg’s conjecture) and its generalization by Sarnak and Xue [47] (see [29] and [26] for related results), asserting that the first eigenvalue of the congruenc... |

40 |
The spectral geometry of a tower of coverings
- Brooks
- 1986
(Show Context)
Citation Context ...taining to parts [a] and[b]). Then, using the fact that (Γ, O) describes S O (Γ, O) as an orbifold covering, one transfers this information to open surfaces S O (Γ, O), using the results of Brooks in =-=[8, 9]-=-. One then transfers the desired property to the surfaces S C (Γ, O) by using Ahlfors-Schwarz - Wolpert Lemma as developed by Brooks in [10, 11] and extended by Brooks and Makover in [12, 13, 14]. The... |

40 |
Distribution functions for largest eigenvalues and their applications
- Tracy, Widom
- 2002
(Show Context)
Citation Context ...essed in terms of certain Painlevé II function and are now believed to describe new universal limit laws for a wide variety of process arising in mathematical physics and interacting particle systems =-=[51]-=-. A dramatic consequence of Conjecture 1 would be that the probability of random regular graph being Ramanujan approaches 0.52 as the size of the graph tends to infinity, corresponding to the skewness... |

25 |
Fuchsian groups, coverings of Riemann surfaces, subgroup growth, random quotients and random walks, preprint
- Liebeck, Shalev
(Show Context)
Citation Context ...Memorial Conference Alex Lubotzky told me that Martin Liebeck and Aner Shalev independently obtained a result similar to Proposition 2; I would like to thank Aner Shalev for sending me their preprint =-=[31]-=-. I would like to thank Persi Diaconis, Eran Makover, Anatoly Vershik, Ofer Zeitouni, and Martin Zerner for interest in this work and stimulating discussions. 2. Belyi surfaces In [4] Belyi proved a r... |

23 | Random Construction of Riemann Surfaces
- Brooks, Makover
(Show Context)
Citation Context ...ann surfaces originated in the work of Buser [15, 17]. As we discuss in Section 3, the behavior of the first eigenvalue of the discrete Laplacian on a random cubic graph is understood rather well. In =-=[14]-=-, Brooks and Makover introduced an approach to studying the first eigenvalue of the Laplacian of a “typical” compact Riemann surface of large genus based on compactifying finite-area Riemann surfaces ... |

21 | Platonic Surfaces
- Brooks
- 1986
(Show Context)
Citation Context ... open surfaces S O (Γ, O), using the results of Brooks in [8, 9]. One then transfers the desired property to the surfaces S C (Γ, O) by using Ahlfors-Schwarz - Wolpert Lemma as developed by Brooks in =-=[10, 11]-=- and extended by Brooks and Makover in [12, 13, 14]. The topology of the surface can be read of from (Γ, O), using LHT paths. In particular, the genus is given by Euler’s formula; as is customary when... |

21 |
What is an expander
- Sarnak
(Show Context)
Citation Context ...aph G is an analogue of the Cheeger’s constant for Riemann surfaces and is defined as follows { (2) c(G) = inf c(X) | |X| < 1 2 |G| } . A family of k-regular graphs Xn,k forms a family of C-expanders =-=[32, 44]-=- if there is a fixed positive constant C, such that (3) lim inf n→∞ c(Xn,k) ≥ C.POISSON-DIRICHLET DISTRIBUTION FOR RANDOM BELYI SURFACES 9 The adjacency matrix of G, A(G) is the |G | by |G | matrix, ... |

21 |
Asymptotics of the largest and the typical dimensions of the irreducible representations of a symmetric group”, Funktsional’nyi Analiz i
- Vershik, Kerov
- 1985
(Show Context)
Citation Context ...> 0 we have ∑ (20) λ⊢N λ1,λ ′ 1 ≤N−m (f λ ) −t = O(N −mt ), where implied constant depends only on m. We remark that the sums of the form ∑ λ⊢N (fλ) β for β > 0 have been studied by Vershik and Kerov =-=[52]-=- and by Regev [43]. In particular, Regev relates the asymptotic computations of such sums to the matrix integral of the form ∫ ∞ ∫ ∞ ∏ . . . −∞ |xi − xj| −∞ i<j β e −(β/2)(x2 1 +···+x2 N ) dx1 . . .dx... |

21 |
Topological characteristics of random triangulated surfaces, Random Structures Algorithms
- Pippenger, Schleich
(Show Context)
Citation Context ...y [49, 57, 58]—to prove the conjecture of Brooks and Makover. It turns out that the number of oriented cycles in random cubic graphs with random orientation was also studied by Pippenger and Schleich =-=[43]-=- in connection with topological characteristics of random surfaces generated by cubic interactions. The surfaces considered by Pippenger and Schleich are obtained by taking 3n arcs of an even number o... |

20 |
Symmetric functions and random partitions, Symmetric functions 2001: surveys of developments and perspectives
- Okounkov
- 2002
(Show Context)
Citation Context ...of the form ∫ ∞ ∫ ∞ ∏ . . . −∞ |xi − xj| −∞ i<j β e −(β/2)(x2 1 +···+x2 N ) dx1 . . .dxN, this being perhaps the first hint of the deep connection between random matrices and random permutations; see =-=[39]-=- and references therein for a recent survey. Proof of Proposition 2: First of all we observe that since fλ = fλ′ , it suffices to prove proposition 2 for the sum ∑ λ⊢N λ ′ 1 <λ1≤N−m Now we split this ... |

15 | The Poisson-Dirichlet law is the unique invariant distribution for uniform split-merge transformations
- Diaconis, Mayer-Wolf, et al.
(Show Context)
Citation Context ...ribution is called Poisson-Dirichlet distribution with parameter θ. Poisson-Dirichlet distribution arises in a great variety of problems, see [3] and references therein; in a recent breakthrough work =-=[20]-=- it wasPOISSON-DIRICHLET DISTRIBUTION FOR RANDOM BELYI SURFACES 3 proved that it is the unique invariant distribution for uniform splitmerge transformations. As we discuss in section 3, the distribut... |

14 | Cubic graphs and the first eigenvalue of a Riemann surface - Buser - 1978 |

13 | Riemann Surfaces with Large First Eigenvalue
- Brooks, Makover
(Show Context)
Citation Context ... Brooks in [8, 9]. One then transfers the desired property to the surfaces S C (Γ, O) by using Ahlfors-Schwarz - Wolpert Lemma as developed by Brooks in [10, 11] and extended by Brooks and Makover in =-=[12, 13, 14]-=-. The topology of the surface can be read of from (Γ, O), using LHT paths. In particular, the genus is given by Euler’s formula; as is customary when using Euler’s formula we refer LHT path as a face ... |

13 |
On the bipartition of graphs
- Buser
- 1984
(Show Context)
Citation Context ... The author was supported in part by the NSF postdoctoral fellowship. 12 ALEX GAMBURD The idea of using cubic graphs to study the first eigenvalue of Riemann surfaces originated in the work of Buser =-=[15, 16]-=-. As we discuss in section 3, one understands rather well the behavior of the first eigenvalue of the discrete Laplacian on a random cubic graph. In [14], Brooks and Makover introduced an approach to ... |

12 |
Random walks on groups: characters and geometry
- Diaconis
- 2003
(Show Context)
Citation Context ...rollary is completed by applying theorem 3 and triangle inequality. We now turn to the proof of theorem 3. The basic tool is the following result, known as Diaconis-Shahshahani upper-bound lemma; see =-=[19]-=- for a recent survey of its applications and ramifications. Proposition 1 (Diaconis-Shahshahani [21]). Let G be a finite group and denote by ˆ G the set of irreducible unitary representations of G. Le... |

11 | Rapidly mixing random walks and bounds on characters of the symmetric groups - Lulov, Pak |

11 |
Small eigenvalues of the Laplace operator on compact Riemann surfaces
- Randol
- 1974
(Show Context)
Citation Context ...25] for related results), asserting that the first eigenvalue of the congruence surfaces of arbitrary genus is bounded away from zero; on the other hand there are examples due to Selberg [46], Randol =-=[42]-=-, and Buser [17], showing that, in general, the first eigenvalue can be made arbitrarily small. The surfaces in the latter examples are “long and thin”, so in some sense live on the boundary of Teichm... |

9 |
Exceptional eigenvalues and congruence subgroups
- Huxley
- 1986
(Show Context)
Citation Context ...rences therein. On the one hand, we have a celebrated theorem of Selberg [46] (see [34] and [30] for refined estimates towards Selberg’s conjecture) and its generalization by Sarnak and Xue [45] (see =-=[27]-=- and [25] for related results), asserting that the first eigenvalue of the congruence surfaces of arbitrary genus is bounded away from zero; on the other hand there are examples due to Selberg [46], R... |

7 | On the genus of a random riemann surface
- Gamburd, Makover
- 2002
(Show Context)
Citation Context ...3] and we will pursue the exhaustive exploitation of the consequences of Theorem 3 in a subsequent publication. Here we note just two immediate corollaries which substantially refine results in [14], =-=[26]-=-, and [41]. Corollary 2. Let l(n) denote the number of LHT paths in a random cubic graph on n vertices with random orientation. Then as n → ∞ E(l(n)) = log(3n) + γ + O(n −1/6 ), Var(l(n)) = log(3n) + ... |

7 |
Stationary random partitions of a natural series. Teor. Veroyatnost. i Primenen., 44(1):55–73
- Tsilevich
- 1999
(Show Context)
Citation Context ...42] and references therein. In i=1POISSON–DIRICHLET DISTRIBUTION FOR RANDOM BELYI SURFACES 3 a recent breakthrough work [21], Diaconis, Mayer-Wolf, Zeitouni and Zerner proved a conjecture of Vershik =-=[52]-=- asserting that Poisson–Dirichlet distribution is the unique invariant distribution for uniform split-merge transformations. As we discuss in Section 3, the distribution of cycle lengths for random re... |

6 | Some Geometric Aspects of the Work of Lars Ahlfors
- Brooks
(Show Context)
Citation Context ...pen surfaces S O (Γ, O), using the results of Brooks in [8, 9]. One then transfers the desired property to the surfaces S C (Γ, O) by using the Ahlfors–Schwarz–Wolpert lemma as developed by Brooks in =-=[10, 11]-=- and extended by Brooks and Makover in [12, 13, 14]. The topology of the surface can be read off from (Γ, O), using LHT paths. In particular, the genus is given by Euler’s formula; as is customary whe... |

6 | Symmetric functions and random partitions
- Okounkov
- 2003
(Show Context)
Citation Context ...of the form ∫ ∞ ∫ ∞ ∏ · · · −∞ |xi − xj| −∞ i<j β e −(β/2)(x2 1 +···+x2 N ) dx1 · · ·dxN, this being one of the first hints of the deep connection between random matrices and random permutations; see =-=[40]-=- and references therein for a recent survey. Proof of Proposition 4.2. First, we observe that since fλ = fλ′ , it suffices to prove Proposition 4.2 for the sum ∑ λ⊢N λ ′ 1 <λ1≤N−m Now we split this su... |

5 |
Some remarks on volume and diameter of Riemannian manifolds
- Brooks
- 1988
(Show Context)
Citation Context ...taining to parts [a] and[b]). Then, using the fact that (Γ, O) describes S O (Γ, O) as an orbifold covering, one transfers this information to open surfaces S O (Γ, O), using the results of Brooks in =-=[8, 9]-=-. One then transfers the desired property to the surfaces S C (Γ, O) by using Ahlfors-Schwarz - Wolpert Lemma as developed by Brooks in [10, 11] and extended by Brooks and Makover in [12, 13, 14]. The... |

5 |
Asymptotic Behavior of the Random 3-Regular Bipartite Graph
- Novikoff
- 2002
(Show Context)
Citation Context ...dded ball in SC (Γ, O) of the total surface area. converges to 0.62 2π In fact, the limiting distribution of L is also known, but we do not pursue it here. Recent numerical experiments of T. Novikoff =-=[38]-=- present convincing evidence in favor of the following conjecture. Conjecture 1. The distribution of the second largest eigenvalue of the adjacency matrix of random regular graph, suitably rescaled, f... |

5 |
Discrete Groups, Expanding Graphs and Invariant
- Lubotzky
- 1994
(Show Context)
Citation Context ...is an analogue8 A. GAMBURD of Cheeger’s constant for Riemann surfaces and is defined as follows: c(G) = inf{c(X)||X| < 1 2 |G|}. (3.1) A family of k-regular graphs Xn,k forms a family of C-expanders =-=[34, 46]-=- if there is a fixed positive constant C such that (3.2) liminf n→∞ c(Xn,k) ≥ C. The adjacency matrix of G, A(G), is the |G| by |G| matrix, with rows and columns indexed by vertices of G, such that th... |

4 |
On Cheeger’s inequality λ1 ≥ h 2 /4
- Buser
- 1979
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Citation Context ... results), asserting that the first eigenvalue of the congruence surfaces of arbitrary genus is bounded away from zero; on the other hand there are examples due to Selberg [48], Randol [44] and Buser =-=[16]-=-, showing that, in general, the first eigenvalue can be made arbitrarily small. The surfaces in the latter examples are “long and thin,” so in some sense live on the boundary of Teichmüller spaces, an... |

4 |
Expander graphs, random matrices and quantum chaos, Random walks and geometry
- Gamburd
- 2004
(Show Context)
Citation Context ...nd Lloyd [49], who also computed the limiting distribution of L.20 A. GAMBURD 5.2. Recent numerical experiments of Novikoff [39] present convincing evidence in favor of the following conjecture (see =-=[27]-=- for a discussion of related conjectures and numerical results): Conjecture 5.1 ([39]). The distribution of the second largest eigenvalue of the adjacency matrix of a random regular graph, suitably re... |

4 |
Limit measures that arise in the asymptotic theory of symmetric groups
- Vershik, Shmidt
- 1977
(Show Context)
Citation Context ...istribution. We then invoke what is perhaps the oldest occurrence of Poisson–Dirichlet distribution— the distribution of normalized cycle lengths for a random permutation in Sn as n tends to infinity =-=[49, 57, 58]-=-—to prove the conjecture of Brooks and Makover. It turns out that the number of oriented cycles in random cubic graphs with random orientation was also studied by Pippenger and Schleich [43] in connec... |

3 |
On Cheeger’s inequality λ1
- Buser
- 1980
(Show Context)
Citation Context ...results), asserting that the first eigenvalue of the congruence surfaces of arbitrary genus is bounded away from zero; on the other hand there are examples due to Selberg [46], Randol [42], and Buser =-=[17]-=-, showing that, in general, the first eigenvalue can be made arbitrarily small. The surfaces in the latter examples are “long and thin”, so in some sense live on the boundary of Teichmüller spaces, an... |

2 | Belyi surfaces
- BROOKS, MAKOVER
- 2001
(Show Context)
Citation Context ... Brooks in [8, 9]. One then transfers the desired property to the surfaces S C (Γ, O) by using Ahlfors-Schwarz - Wolpert Lemma as developed by Brooks in [10, 11] and extended by Brooks and Makover in =-=[12, 13, 14]-=-. The topology of the surface can be read of from (Γ, O), using LHT paths. In particular, the genus is given by Euler’s formula; as is customary when using Euler’s formula we refer LHT path as a face ... |

1 |
Logarithmic combinatorial structrues: a probabilistic approach
- Arratia, Barbour, et al.
- 2003
(Show Context)
Citation Context ...with random orientation; Brooks and Makover conjectured that asymptotically normalized cycles lengths follow Poisson-Dirichlet distribution. We recall the definition of Poisson-Dirichlet distribution =-=[3]-=-. Let B1, B2, . . . be independent random variables uniformly distributed on [0, 1]. Define G = (G1, G2, . . ...) as follows: G1 = B1; G2 = (1−B1)B2; . . .Gi = (1−B1)(1−B2) . . .(1−Bi−1)Bi. The random... |

1 |
Some geometric aspects of the work
- Brooks
(Show Context)
Citation Context ... open surfaces S O (Γ, O), using the results of Brooks in [8, 9]. One then transfers the desired property to the surfaces S C (Γ, O) by using Ahlfors-Schwarz - Wolpert Lemma as developed by Brooks in =-=[10, 11]-=- and extended by Brooks and Makover in [12, 13, 14]. The topology of the surface can be read of from (Γ, O), using LHT paths. In particular, the genus is given by Euler’s formula; as is customary when... |

1 |
On the number of rim hook tableux
- Fomin, Lulov
- 1997
(Show Context)
Citation Context ...ing of the product of 2-cycles in the symmetric group SN with N = nk. In section 4 using Diaconis-Shahshahani upper-bound lemma [21], the estimate on the number of rim hook tableux by Fomin and Lulov =-=[22]-=-, and representation theory of the symmetric group (in particular, hook-length formula and Murnaghan-Nakayama rule), we show that as n → ∞ the distribution of βα converges to uniform distribution. We ... |

1 |
On Selberg’s eigenvalue conjecture, GAFA 5
- Luo, Rudnick, et al.
- 1995
(Show Context)
Citation Context ...iemann surfaces of increasing genus has received considerable attention over the last thirty years; see [18] and references therein. On the one hand, we have a celebrated theorem of Selberg [46] (see =-=[34]-=- and [30] for refined estimates towards Selberg’s conjecture) and its generalization by Sarnak and Xue [45] (see [27] and [25] for related results), asserting that the first eigenvalue of the congruen... |

1 |
Topological characterisitcs of random surfaces generated by cubic interactions
- Pippenger, Schleich
(Show Context)
Citation Context ... as n tends to infinity – to prove Brooks-Makover conjecture. It turns out that the number of oriented cycles in random cubic graphs with random orientation was also studied by Pippenger and Schleich =-=[41]-=- in connection with topological characteristics of random surfaces generated by cubic interactions. (I am grateful to Bálint Virág for bringing the paper [41] to my attention). The surfaces considered... |

1 |
A partition function connected with Young diagrams
- Vershik
- 1989
(Show Context)
Citation Context ... ∑ (4.13) λ⊢N λ1,λ ′ 1 ≤N−m (f λ ) −t = O(N −mt ), where the implied constant depends only on m. We remark that the sums of the form ∑ λ⊢N (fλ) β for β > 0 have been studied by Regev [45] and Vershik =-=[56]-=-. In particular, Regev relates the asymptotic computations of such sums to the matrix integral of the form ∫ ∞ ∫ ∞ ∏ · · · −∞ |xi − xj| −∞ i<j β e −(β/2)(x2 1 +···+x2 N ) dx1 · · ·dxN, this being one ... |