Semi-classical limit for random walks

Ursula Porod*, Steve Zelditch

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


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 L2(G). The irreducibles Vp of the left regular representation of G on L2(G) are finite dimensional invariant subspaces for Tμ and the spectrum of Tμ is the union of the sub-spectra simg;a(Tμ 1vρ) on the irreducibles, which consist of real eigenvalues {λρ1,..., λρdim;vρ}. 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 Tμ, is essentially a direct sum of Harper operators).

Original languageEnglish (US)
Pages (from-to)5317-5355
Number of pages39
JournalTransactions of the American Mathematical Society
Issue number11
StatePublished - 2000

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics


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