The Annals of Probability

The cut-off phenomenon for random reflections

Ursula Porod

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For many random walks on "sufficiently large" finite groups the so-called cut-off phenomenon occurs: roughly stated, there exists a number $k_0$ , depending on the size of the group, such that $k_0$ 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 $k_0 = 1/2 N\log N$.

Article information

Ann. Probab., Volume 24, Number 1 (1996), 74-96.

First available in Project Euclid: 15 January 2003

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J15 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization

Random walk reflection cut-off phenomenon Fourier analysis


Porod, Ursula. The cut-off phenomenon for random reflections. Ann. Probab. 24 (1996), no. 1, 74--96. doi:10.1214/aop/1042644708.

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