## Eigenvalues of Random Wreath Products (2002)

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Venue: | Electron. J. Probab |

Citations: | 3 - 0 self |

### BibTeX

@ARTICLE{Evans02eigenvaluesof,

author = {Steven N. Evans},

title = {Eigenvalues of Random Wreath Products},

journal = {Electron. J. Probab},

year = {2002},

volume = {7},

pages = {pp. (electronic).}

}

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### Abstract

. Consider a unifomrly chosen element Xn of the n-fold wreath product \Gamma n = G o G o \Delta \Delta \Delta o G, where G is a finite permutation group acting transitively on some set of size s. The eigenvalues of Xn in the natural s n - dimensional permutation representation (the composition representation) are investigated by considering the random measure \Xi n on the unit circle that assigns mass 1 to each eigenvalue. It is shown that if f is a trigonometric polynomial, then limn!1 Pf R f d\Xi n 6= s n R f dg = 0, where is normalised Lebesgue measure on the unit circle. In particular, s \Gamman \Xi n converges weakly in probability to as n ! 1. For a large class of test functions f with nonterminating Fourier expansions, it is shown that there exists a constant c and a non-zero random variable W (both depending on f) such that c \Gamman R f d\Xi n converges in distribution as n !1 to W . These results have applications to Sylow p-groups of symmetric groups and autmorphism groups of regular rooted trees. 1.

### Citations

652 |
Random Matrices
- Mehta
- 1967
(Show Context)
Citation Context ...m matrices in general and eigenvalues of Haar distributed random matrices from various compact groups in particular. A general reference to the history of random matrix theory and its applications is =-=[Meh91]-=-. Asymptotics for the traces of powers of unitary, orthogonal and symplectic matrices (equivalently, integrals of powers against the analogue of the measure \Xi n ) are investigated in [DS94] (see als... |

428 |
A.Kerber, The representation theory of the symmetric group, Addison-Wesley
- James
- 1981
(Show Context)
Citation Context ...the r-fold wreath product C p o C p o \Delta \Delta \Delta o C p , where C p is the cyclic group of order p. The Sylow p-group of Sn for a general n is a certain product of such groups (see 4.1.22 of =-=[JK81]-=-). For G and H as above, there is a natural representation of G o H as a group of permutations of the set f0; 1; : : : ; t \Gamma 1g \Theta f0; 1; : : : ; s \Gamma 1g. In this permutation representati... |

109 | Random walks on discrete groups: boundary and - Kaı̆manovich, Vershik - 1983 |

75 |
On the eigenvalues of random matrices
- Diaconis, Shahshahani
- 1994
(Show Context)
Citation Context ...ions is [Meh91]. Asymptotics for the traces of powers of unitary, orthogonal and symplectic matrices (equivalently, integrals of powers against the analogue of the measure \Xi n ) are investigated in =-=[DS94]-=- (see also [Rai97]). Integrals of more general well-behaved 10 STEVEN N. EVANS functions against the analogue of \Xi n for these groups are studied in [Joh97]. The number of eigenvalues in an interval... |

55 | Linear functionals of eigenvalues of random matrices
- Diaconis, Evans
- 2001
(Show Context)
Citation Context ...nst the analogue of \Xi n , and this object is the subject of [HKO00, KS00a, KS00b]. A general theory for the unitary, orthogonal and symplectic groups that subsumes much of this work is presented in =-=[DE01]-=-. Random permutations give rise to random permutation matrices. Given the connection between cycle counts of permutations and traces of the corresponding matrices, some of the huge literature on the c... |

42 | On the characteristic polynomial of a random unitary matrix - Hughes, Keating, et al. - 2001 |

35 |
On random matrices from the compact classical groups
- Johansson
- 1997
(Show Context)
Citation Context ...of the measure \Xi n ) are investigated in [DS94] (see also [Rai97]). Integrals of more general well-behaved 10 STEVEN N. EVANS functions against the analogue of \Xi n for these groups are studied in =-=[Joh97]-=-. The number of eigenvalues in an interval for the unitary group (that is, the integral of an indicator function against the analogue of \Xi n ) is investigated in [Wie98]. The logarithm of the charac... |

33 | Random walks on the lamplighter group - Lyons, Pemantle, et al. - 1993 |

31 | C.: Random matrix theory and L-functions at s = 1/2 - Keating, Snaith - 2000 |

22 |
High powers of random elements of compact Lie groups. Probab. Theory Related Fields 107
- Rains
- 1997
(Show Context)
Citation Context ...Asymptotics for the traces of powers of unitary, orthogonal and symplectic matrices (equivalently, integrals of powers against the analogue of the measure \Xi n ) are investigated in [DS94] (see also =-=[Rai97]-=-). Integrals of more general well-behaved 10 STEVEN N. EVANS functions against the analogue of \Xi n for these groups are studied in [Joh97]. The number of eigenvalues in an interval for the unitary g... |

16 |
Eigenvalue Distributions of Random Matrices in the Permutation Group and Compact Lie Groups
- Wieand
- 1998
(Show Context)
Citation Context ... these groups are studied in [Joh97]. The number of eigenvalues in an interval for the unitary group (that is, the integral of an indicator function against the analogue of \Xi n ) is investigated in =-=[Wie98]-=-. The logarithm of the characteristic polynomial of a random unitary matrix is also the integral of a suitable function against the analogue of \Xi n , and this object is the subject of [HKO00, KS00a,... |

15 | Representations of Permutation Groups I - Kerber - 1971 |

12 |
Generalized wreath products of permutation groups
- Bailey, Praeger, et al.
- 1983
(Show Context)
Citation Context ...ld wreath product Cp oCp o oCp, where Cp is the cyclic group of order p (see 4.1.22 of [JK81]). A review of the uses of wreath products in experimental design and the analysis of variance is given in =-=[BPRS83]-=-. An intriguing application to spectroscopy can be found in [LH63]. For G and H as above, there is a natural representation of G o H as a group of permutations of the set f0; 1;:::;t, 1g f0; 1;:::;s, ... |

10 |
Renormalization and automorphism groups of rooted trees
- Bass, Otero-Espinar, et al.
(Show Context)
Citation Context ...ci cally with the automorphism groups of regular rooted trees and their representation theory is [AD02]. A connection between automorphism groups of regular trees and dynamical systems is explored in =-=[BOERT96]-=-. Besides their appearance as the automorphism groups of regular rooted trees, wreath products are important in the representation theory of the symmetric group and in various problems arising in the ... |

9 |
Eigenvalue Distributions of Random Permutation Matrices, The Annals of Probability 28
- Wieand
- 2005
(Show Context)
Citation Context ...nts about eigenvalues of random permutation matrices. More in the spirit of this paper, the number of eigenvalues in an interval and the logarithm of the characteristic polynomial are investigated in =-=[Wie00]-=- and [HKOS00], respectively. The former paper treats not only the symmetric group, but also the wreath product of a cyclic group with a symmetric group. There is a limited literature on other probabil... |

7 | Characteristics of random walks on wreath products of groups - Dyubina |

7 | The characteristic polynomial of a random permutation matrix
- Hambly, Keevash, et al.
- 2000
(Show Context)
Citation Context ...genvalues of random permutation matrices. More in the spirit of this paper, the number of eigenvalues in an interval and the logarithm of the characteristic polynomial are investigated in [Wie00] and =-=[HKOS00]-=-, respectively. The former paper treats not only the symmetric group, but also the wreath product of a cyclic group with a symmetric group. There is a limited literature on other probabilistic aspects... |

5 | Random matrix theory and i(1/2 - Keating, Snaith - 2001 |

4 | Mixing times for Markov chains on wreath products and related homogeneous spaces, Electron
- Fill, Schoolfield
(Show Context)
Citation Context ...studied in [PS83a, PS89]. The probability that a randomly chosen element of Sn o S p has no fixed points as n !1 is given in [DS95]. Mixing times of Markov chains on wreath products are considered in =-=[FS01]-=-. Finally, infinite wreath products are a fruitful source of examples of interesting behaviour and counterexamples in the study of random walks on infinite groups (see, for example, [KV83, LPP96, PSC9... |

4 |
On a problem of P. Turán concerning Sylow subgroups
- Pálfy, Szalay
- 1983
(Show Context)
Citation Context ...re on other probabilistic aspects of wreath products. As mentioned above, the Sylow p-group of S p r is a wreath product. The distribution of the order of a random element of this group is studied in =-=[PS83b]-=-, while the distribution of the degree of a randomly chosen irreducible character is studied in [PS83a, PS89]. The probability that a randomly chosen element of Sn o S p has no fixed points as n !1 is... |

4 | Random matrix theory and (1=2+it - Keating, Snaith - 2000 |

3 | Amenable groups, isoperimetric profiles and random walks - Saloff-Coste - 1996 |

3 |
The symmetry groups of non-rigid molecules
- Longuet-Higgins
- 1963
(Show Context)
Citation Context ...r p (see 4.1.22 of [JK81]). A review of the uses of wreath products in experimental design and the analysis of variance is given in [BPRS83]. An intriguing application to spectroscopy can be found in =-=[LH63]-=-. For G and H as above, there is a natural representation of G o H as a group of permutations of the set f0; 1;:::;t, 1g f0; 1;:::;s, 1g. In this permutation representation, the group element(f; ) 2 G... |

2 | An example of the rate of departure to infinity for a random walk on a group - DYUBINA - 1999 |

2 | Random walks on discrete groups: boundary and entropy - manovich, A, et al. - 1983 |

1 |
The (generalized) secretary's packet problem and the Bell numbers
- Dale, Skau
- 1995
(Show Context)
Citation Context ...e distribution of the degree of a randomly chosen irreducible character is studied in [PS83a, PS89]. The probability that a randomly chosen element of Sn o S p has no fixed points as n !1 is given in =-=[DS95]-=-. Mixing times of Markov chains on wreath products are considered in [FS01]. Finally, infinite wreath products are a fruitful source of examples of interesting behaviour and counterexamples in the stu... |

1 | The distribution of the character degrees of the symmetric p-groups - P'alfy, Szalay - 1983 |

1 | Further probabilistic results on the symmetric p-groups - P'alfy, Szalay - 1989 |

1 |
One{two{tree(s) and Sylow subgroups of Sn
- Abert, Diaconis
- 2002
(Show Context)
Citation Context ... for wreath products with extensive bibliographies are [Ker71, Ker75, JK81]. A reference that deals speci cally with the automorphism groups of regular rooted trees and their representation theory is =-=[AD02]-=-. A connection between automorphism groups of regular trees and dynamical systems is explored in [BOERT96]. Besides their appearance as the automorphism groups of regular rooted trees, wreath products... |

1 |
Groups acting on regular trees: probability and Hausdor dimension
- Virag
- 2002
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Citation Context ...metric group. There is a limited literature on other probabilistic aspects of wreath products. Various connections between automorphism groups of in nite regular trees and probability are explored in =-=[AV02]-=-. As we recall in Section 2, the Sylow p-group of Spr is a wreath product. The distribution of the order of a random element of this group is studied in [PS83b], while the distribution of the degree o... |

1 | An example of the rate of departure to in nity for a random walk on a group - Dyubina - 1999 |

1 | The distributionofthecharacter degrees of the symmetric p-groups - Palfy, Szalay - 1983 |

1 | Amenable groups, isoperimetric pro les and random walks - Pittet, -Coste - 1996 |