## Abstract

Let (G,μ) be a discrete symmetric random walk on a compact Lie group G with step distribution μ and let Tμ be the associated transition operator on L^{2}(G). The irreducibles V_{p} of the left regular representation of G on L^{2}(G) are finite dimensional invariant subspaces for T_{μ} and the spectrum of T_{μ} is the union of the sub-spectra simg;a(T_{μ} 1v_{ρ}) on the irreducibles, which consist of real eigenvalues {λ_{ρ}1,..., λ_{ρdim;}v_{ρ}}. Our main result is an asymptotic expansion for the spectral measures Matrix Equation along rays of representations in a positive Weyl chamber t., i.e. for sequences of representations kp, k ∈ N with k Ρ ∞. As a corollary we obtain some estimates on the spectral radius of the random walk. We also analyse the fine structure of the spectrum for certain random walks on U(n) (for which T_{μ}, is essentially a direct sum of Harper operators).

Original language | English (US) |
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Pages (from-to) | 5317-5355 |

Number of pages | 39 |

Journal | Transactions of the American Mathematical Society |

Volume | 352 |

Issue number | 11 |

DOIs | |

State | Published - 2000 |

## ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics