The cut-off phenomenon for random reflections

Ursula Porod*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

23 Scopus citations

Abstract

For many random walks on "sufficiently large" finite groups the so-called cut-off phenomenon occurs: roughly stated, there exists a number k0, depending on the size of the group, such that k0 steps are necessary and sufficient for the random walk to closely approximate uniformity. As a first example on a continuous group, Rosenthal recently proved the occurrence of this cut-off phenomenon for a specific random walk on SO(N). Here we present and [for the case of O(N)] prove results for random walks on O(N), U(N) and Sp(N), where the one-step distribution is a suitable probability measure concentrated on reflections. In all three cases the cut-off phenomenon occurs at k0 = 1/2N log N.

Original languageEnglish (US)
Pages (from-to)74-96
Number of pages23
JournalAnnals of Probability
Volume24
Issue number1
DOIs
StatePublished - 1996

Keywords

  • Cut-off phenomenon
  • Fourier analysis
  • Random walk
  • Reflection

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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