## Abstract

For each n ≥ 2, let A _{n} = (ξ _{ij}) be an n × n symmetric matrix with diagonal entries equal to zero and the entries in the upper triangular part being independent with mean μ _{n} and standard deviation σ _{n}. The Laplacian matrix is defined by. In this paper, we obtain the laws of large numbers for λ _{n-k}(Δ _{n}), the (k + 1)-th smallest eigenvalue of Δ _{n}, through the study of the order statistics of weakly dependent random variables. Under certain moment conditions on ξ _{ij}'s, we prove that, as n → ∞, for any k ≥ 1. Further, if {Δ _{n}; n ≥ 2} are independent with μ _{n} = 0 and σ _{n} = 1, then, (ii) the sequence, is dense in, for any k ≥ 0. In particular, (i) holds for the Erdös-Rényi random graphs. Similar results are also obtained for the largest eigenvalues of Δ _{n}.

Original language | English (US) |
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Pages (from-to) | 671-690 |

Number of pages | 20 |

Journal | Probability Theory and Related Fields |

Volume | 153 |

Issue number | 3-4 |

DOIs | |

State | Published - Aug 1 2012 |

## Keywords

- Extreme eigenvalues
- Laplacian matrix
- Order statistics
- Random graph
- Random matrix