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

Venue: | Ann. Probab |

Citations: | 4 - 0 self |

### BibTeX

@ARTICLE{Gamburd06poisson-dirichletdistribution,

author = {Alex Gamburd},

title = {Poisson-Dirichlet distribution for random Belyi surfaces},

journal = {Ann. Probab},

year = {2006}

}

### OpenURL

### Abstract

Abstract. 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 cycles lengths follow Poisson-Dirichlet distribution. We present a proof of this conjecture using representation theory of the symmetric group. Consequently we also make progress towards a conjecture of Pippenger and Schleich which arose in the study of topological characteristics of random surfaces generated by cubic interactions. 1.

### Citations

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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... |

413 |
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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... |

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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... |

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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 ... |

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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... |

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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... |

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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 [... |

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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... |

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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... |

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Combinatorial stochastic processes
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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... |

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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... |

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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... |

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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... |

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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... |

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Galois extensions of a maximal cyclotomic field
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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... |

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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... |

63 | On the second eigenvalue and random walks in random regular graphs
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- 1991
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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... |

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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 . ... |

59 | On the complexity of a concentrator, The
- Pinsker
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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... |

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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... |

50 |
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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, ... |

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Models of random regular graphs, Surveys in Combinatorics
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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... |

47 |
The isoperimetric number of random regular graphs
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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... |

34 | Ribbon graphs, quadratic differentials on Riemann surfaces, and algebraic curves defined over Q, Asian
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- 1998
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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... |

34 |
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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... |

29 |
The spectral geometry of a tower of coverings
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- 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... |

28 |
Appendix: Refined estimates towards the Ramanujan and Selberg conjectures
- Kim, Sarnak
(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... |

26 |
Xue: Bounds for multiplicities of automorphic representations
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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... |

26 |
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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... |

23 |
Distribution functions for largest eigenvalues and their applications
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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... |

21 |
On the spectral gap for infinite index “congruence” subgroups of SL2(Z
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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... |

19 | Belyĭ functions, hypermaps, and Galois groups
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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 ... |

17 |
Fuchsian groups, coverings of Riemann surfaces, subgroup growth, random quotients and random walks
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(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... |

15 |
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(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... |

14 | Platonic Surfaces
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- 1999
(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... |

11 | Random construction of Riemann surfaces
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- 1997
(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 ... |

10 | The Poisson-Dirichlet law is the unique invariant distribution for uniform split-merge transformations
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- 2004
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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... |

10 |
Symmetric functions and random partitions, Symmetric functions 2001: surveys of developments and perspectives
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- 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 ... |

9 |
Exceptional eigenvalues and congruence subgroups, in “The Selberg trace formula and related topics”, p. 341– 349; edited by
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- 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... |

9 |
What is an expander
- Sarnak
- 2004
(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, ... |

8 |
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 ... |

8 |
Random walks on groups: Characters and geometry
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- 2000
(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... |

7 | 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 ... |

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

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

7 | 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... |

7 |
K.: Topological characteristics of random triangulated surfaces
- Pippenger, Schleich
- 2006
(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... |

6 |
Asymptotic behavior of the random 3-regular bipartite graph, preprint
- Novikoff
(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... |

6 | Symmetric functions and random partitions
- Okounkov
(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... |