## Eigenvalue spacings for regular graphs (1999)

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Venue: | IN IMA VOL. MATH. APPL |

Citations: | 10 - 5 self |

### BibTeX

@INPROCEEDINGS{Jakobson99eigenvaluespacings,

author = {Dmitry Jakobson and Stephen D. Miller and Igor Rivin and Zeév Rudnick},

title = {Eigenvalue spacings for regular graphs},

booktitle = {IN IMA VOL. MATH. APPL},

year = {1999},

pages = {317--327},

publisher = {Springer}

}

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### Abstract

We carry out a numerical study of fluctuations in the spectrum of regular graphs. Our experiments indicate that the level spacing distribution of a generic k-regular graph approaches that of the Gaussian Orthogonal Ensemble of random matrix theory as we increase the number of vertices. A review of the basic facts on graphs and their spectra is included.

### Citations

1934 | Random Graphs - Bollobás - 1985 |

183 |
Discrete groups, expanding graphs and invariant measures
- Lubotzky
- 1994
(Show Context)
Citation Context ...hus if we start from an input vertex and walk any even number of steps then we will only be able to land on another input, never on an output. There are examples (such as some Cayley graphs, see [3], =-=[13]-=-) where there are systematic multiplicities in the spectrum and the level spacing distribution at best exists only in some singular limit. For instance in the case of Cayley graphs of the cyclic group... |

157 | Models of random regular graphs
- Wormald
- 1999
(Show Context)
Citation Context ...described in the following section indicate that the same is true for eigenvalues of random regular graphs. 4. Random graph generation We generated random k-regular graphs using a method described in =-=[19]-=-. This method has the virtues of the ease of implementation and of being extremely efficient for the small (≤ 6) values of k of current interest to us. On the other hand, the running time of the algor... |

130 |
Symmetric random walks on groups
- Kesten
- 1959
(Show Context)
Citation Context ...2.4) approach those of the spectral density of the of the infinite k-regular tree T k as n !1. It follows ([15]) that the spectral density (2.4) for a general G 2 G n;k converges to the tree density (=-=[11]-=-) given by f k (x) = 8 ? ! ? : k(4(k \Gamma 1) \Gamma x 2 ) 1=2 2(k 2 \Gamma x 2 ) 0 jxjs2 p k \Gamma 1 jxj ? 2 p k \Gamma 1 (2.5) supported in I k = [\Gamma2 p k \Gamma 1; 2 p k \Gamma 1]. We refer t... |

102 |
Random Matrices, Second Edition
- Mehta
- 1991
(Show Context)
Citation Context ...the statistical model relevant to graphs. It is the space of N \Theta N real symmetric matrices H = (H ij ) with a probability measure P (H)dH which satisfies 3 The standard reference is Mehta's book =-=[16]-=-. EIGENVALUE SPACINGS FOR REGULAR GRAPHS 7 1. P (H)dH is invariant under all orthogonal changes of basis: P (XHX \Gamma1 )dH = P (H)dH; X 2 O(N ) 2. Different matrix elements are statistically indepen... |

74 |
Algebraic Graph Theory. Second Edition
- Biggs
- 1993
(Show Context)
Citation Context ...sa. Thus if we start from an input vertex and walk any even number of steps then we will only be able to land on another input, never on an output. There are examples (such as some Cayley graphs, see =-=[3]-=-, [13]) where there are systematic multiplicities in the spectrum and the level spacing distribution at best exists only in some singular limit. For instance in the case of Cayley graphs of the cyclic... |

60 |
The expected eigenvalue distribution of a large regular graph
- McKay
- 1981
(Show Context)
Citation Context ... G is asymptotic to that of the tree. Accordingly, the r-th moments of the spectral density (2.4) approach those of the spectral density of the of the infinite k-regular tree T k as n !1. It follows (=-=[15]-=-) that the spectral density (2.4) for a general G 2 G n;k converges to the tree density ([11]) given by f k (x) = 8 ? ! ? : k(4(k \Gamma 1) \Gamma x 2 ) 1=2 2(k 2 \Gamma x 2 ) 0 jxjs2 p k \Gamma 1 jxj... |

57 |
clustering in the regular spectrum
- Berry, Tabor
- 1977
(Show Context)
Citation Context ...here the classical motion is the geodesic flow). It has been conjectured that generically there is a remarkable dichotomy: 1. If the classical dynamics are completely integrable, then Berry and Tabor =-=[2]-=- conjectured that the fluctuations are the same as those of an uncorrelated sequence of levels, and in particular P (s) = e \Gammas is Poissonian. 2. If the classical dynamics are chaotic then Bohigas... |

56 | Bollobas Random Graphs - unknown authors - 1985 |

38 |
The asymptotic distribution of short cycles in random regular graphs
- Wormald
- 1981
(Show Context)
Citation Context ... graph G is equal to the number of such closed walks starting and ending at any vertex of the infinite k-regular tree T k . We denote by G n;k the set of k-regular graphs with n vertices. It is known =-=[19] (and not -=-hard to see) that for any fixed rs3 the expected number c r (G) of rcycles in a regular graph G 2 G n;k approaches a constant as n !1; accordingly, for "most" graphs G 2 G n;k c r (G)=n ! 0 ... |

21 |
Chaotic motion and random matrix theories
- Bohigas, Giannoni
- 1984
(Show Context)
Citation Context ...uations are the same as those of an uncorrelated sequence of levels, and in particular P (s) = e \Gammas is Poissonian. 2. If the classical dynamics are chaotic then Bohigas, Giannoni and Schmit [4], =-=[5]-=- conjectured that the fluctuations are modeled by the eigenvalues of a large random symmetric matrix - the Gaussian Orthogonal Ensemble (GOE) 1 . That is, the statistics of the spectral fluctuations a... |

20 |
Chaotic Billiards Generated by Arithmetic Groups
- Bogomolny, Georgeot, et al.
- 1992
(Show Context)
Citation Context ...2k + 1), and more subtle examples in the chaotic case, such as the modular surface (the quotient of the upper halfplane by the modular group SL(2; Z)), where the spacings appear to be Poissonian [1], =-=[6]-=-, [7], [14], there is sufficient numerical evidence for us to believe that these universality conjectures hold in the generic case. In the hope of gaining some extra insight into this matter we checke... |

14 |
Universality of level correlation function of sparse random matrices
- MIRLIN, FYODOROV
- 1991
(Show Context)
Citation Context ...bed in Section 5, indicates that the resulting family of graphs have GOE spacings. This should be compared with the numerical investigations by Evangelou [9] and the discussion by Mirlin and Fyodorov =-=[17]-=- which suggest that in the case of sparse random symmetric matrices the spacings are GOE. We are thus led to conjecture that for a fixed degree ks3, the eigenvalues of the generic k-regular graph on a... |

12 |
Spacing (with discussion
- Pyke
- 1965
(Show Context)
Citation Context ...j e \Gamma2j 2 z 2 In the case that the s i 's are spacings of uncorrelated levels (hence certainly not independent!), the level spacing distribution is exponential P (s) = e \Gammas as N !1 and Pyke =-=[18]-=- derives a limit law for the normalized discrepancy. In the case where the s i 's are spacings of certain models of RMT (not GOE, however) , Katz and Sarnak [10] prove that the discrepancy goes to zer... |

8 |
Energy-Ievel statistics of the Hadamard-Gutzwiller ensemble
- Aurich, Steiner
- 1990
(Show Context)
Citation Context ...city 2k + 1), and more subtle examples in the chaotic case, such as the modular surface (the quotient of the upper halfplane by the modular group SL(2; Z)), where the spacings appear to be Poissonian =-=[1]-=-, [6], [7], [14], there is sufficient numerical evidence for us to believe that these universality conjectures hold in the generic case. In the hope of gaining some extra insight into this matter we c... |

6 |
personal communication
- Lafferty
- 1997
(Show Context)
Citation Context ...= n, so that P (s) = ffi (s \Gamma 1) is a Dirac delta function. Another special example, analogous to the modular surface, seems to have Poisson spacings (numerical evidence by Lafferty and Rockmore =-=[12]-=-). These examples have certain symmetries or degeneracies. We tested a number of families of generic (pseudo)-random k-regular graphs (see section 4 for the details of the generation algorithm). The n... |

5 | The spacing distributions between zeros of zeta functions
- Katz, Sarnak
- 1996
(Show Context)
Citation Context ...ial P (s) = e \Gammas as N !1 and Pyke [18] derives a limit law for the normalized discrepancy. In the case where the s i 's are spacings of certain models of RMT (not GOE, however) , Katz and Sarnak =-=[10]-=- prove that the discrepancy goes to zero almost surely as N ! 1 and conjecture that there is a limit law as in the case of KolmogorovSmirnov and Pyke. Miller (work in progress) has investigated this d... |

2 |
Distribution of Eigenvalues for the Modular
- Bogomolny, Leyvraz, et al.
- 1996
(Show Context)
Citation Context ...1), and more subtle examples in the chaotic case, such as the modular surface (the quotient of the upper halfplane by the modular group SL(2; Z)), where the spacings appear to be Poissonian [1], [6], =-=[7]-=-, [14], there is sufficient numerical evidence for us to believe that these universality conjectures hold in the generic case. In the hope of gaining some extra insight into this matter we checked flu... |

2 |
Lubotzky Discrete Groups, expanding graphs and invariant measures, Birkhauser
- unknown authors
- 1994
(Show Context)
Citation Context ...hus if we start from an input vertex and walk any even number of steps then we will only be able to land on another input, never on an output. There are examples (such as some Cayley graphs, see [3], =-=[13]-=-) where there are systematic multiplicities in the spectrum and the level spacing distribution at best exists only in some singular limit. For instance in the case of Cayley graphs of the cyclic group... |

1 |
A Numerical Study of Sparse Random
- Evangelou
- 1992
(Show Context)
Citation Context ... The numerical evidence we accumulated, described in Section 5, indicates that the resulting family of graphs have GOE spacings. This should be compared with the numerical investigations by Evangelou =-=[9]-=- and the discussion by Mirlin and Fyodorov [17] which suggest that in the case of sparse random symmetric matrices the spacings are GOE. We are thus led to conjecture that for a fixed degree ks3, the ... |

1 | Level Spacings for Cayley Graphs
- Lafferty, Rockmore
(Show Context)
Citation Context ...= n, so that P (s) = ffi (s \Gamma 1) is a Dirac delta function. Another special example, analogous to the modular surface, seems to have Poisson spacings (numerical evidence by Lafferty and Rockmore =-=[12]-=-). These examples have certain symmetries or degeneracies. We tested a number of families of generic (pseudo)-random k-regular graphs (see section 4 for the details of the generation algorithm). The n... |

1 |
Number Variance for Arithmetic Hyperbolic
- Luo, Sarnak
- 1994
(Show Context)
Citation Context ...nd more subtle examples in the chaotic case, such as the modular surface (the quotient of the upper halfplane by the modular group SL(2; Z)), where the spacings appear to be Poissonian [1], [6], [7], =-=[14]-=-, there is sufficient numerical evidence for us to believe that these universality conjectures hold in the generic case. In the hope of gaining some extra insight into this matter we checked fluctuati... |

1 |
Generating random regular graphs, Journal of Algorithms 5
- Wormald
- 1984
(Show Context)
Citation Context ...described in the following section indicate that the same is true for eigenvalues of random regular graphs. 4. Random graph generation We generated random k-regular graphs using a method described in =-=[20]-=-. It is easy to implement and extremely efficient for the small ( 6) values of k of current interest to us. On the other hand, the running time of the algorithm grows exponentially with the degree k, ... |