Source: Ann. Probab.
Volume 24, Number 1
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 $k_0 = 1/2 N\log N$.
1 ALDOUS, D. and DIACONIS, P. 1987. Strong uniform times and finite random walks. Adv. in Appl. Math. 8 69 97.
2 BROCKER, TH. and TOM DIECK, T. 1985. Representations of Compact Lie Groups. Springer, ¨ New York.
3 DIACONIS, P. 1977. Group Representations in Probability and Statistics. IMS, Hay ward, CA.
4 DIACONIS, P. and SHAHSHAHANI, M. 1986. Products of random matrices as they arise in the study of random walks on groups. Contemp. Math. 50 183 195.
5 DIACONIS, P. and SHAHSHAHANI, M. 1987. The subgroup algorithm for generating uniform random variables. Probab. Eng. Inform. Sci. 1 15 32.
6 FULTON, W. and HARRIS, J. 1991. Representation Theory: A First Course. Springer, New York.
7 POROD, U. 1995. L -lower bounds for a speical class of random walks. Probab. Theory 2 Related Fields 101 277 289.
8 POROD, U. 1994. The cut-off phenomenon for random reflections II: complex and quaternionic cases. Preprint.
9 ROSENTHAL, J. S. 1994. Random rotations: characters and random walks on SO N. Ann. Probab. 22 398 423.
11 SLOANE, N. J. A. 1983. Encry pting by random rotations. Cry ptography. Lecture Notes in Computer Science 149 71 128. Springer, New York.
12 ZELOBENKO, D. P. 1973. Compact Lie Groups and Their Representations. AMS Trans. Math. Monographs 40. Amer. Math. Soc., Providence, RI.
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