Coming soon! Project Euclid's new website. Learn more »

### The cut-off phenomenon for random reflections

Ursula Porod
Source: Ann. Probab. Volume 24, Number 1 (1996), 74-96.

#### Abstract

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 $k_0 = 1/2 N\log N$.

First Page:
Primary Subjects: 60J15, 60B15
Full-text: Open access

Permanent link to this document: http://projecteuclid.org/euclid.aop/1042644708
Mathematical Reviews number (MathSciNet): MR1387627
Digital Object Identifier: doi:10.1214/aop/1042644708

### References

1 ALDOUS, D. and DIACONIS, P. 1987. Strong uniform times and finite random walks. Adv. in Appl. Math. 8 69 97.
Mathematical Reviews (MathSciNet): MR88d:60175
Zentralblatt MATH: 0631.60065
Digital Object Identifier: doi:10.1016/0196-8858(87)90006-6
2 BROCKER, TH. and TOM DIECK, T. 1985. Representations of Compact Lie Groups. Springer, ¨ New York.
Mathematical Reviews (MathSciNet): MR86i:22023
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.
Mathematical Reviews (MathSciNet): MR87k:60025
Zentralblatt MATH: 0586.60012
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.
Mathematical Reviews (MathSciNet): MR93a:20069
7 POROD, U. 1995. L -lower bounds for a speical class of random walks. Probab. Theory 2 Related Fields 101 277 289.
Mathematical Reviews (MathSciNet): MR1318197
Zentralblatt MATH: 0815.60006
Digital Object Identifier: doi:10.1007/BF01375829
8 POROD, U. 1994. The cut-off phenomenon for random reflections II: complex and quaternionic cases. Preprint.
Mathematical Reviews (MathSciNet): MR1373375
Zentralblatt MATH: 0865.60005
Digital Object Identifier: doi:10.1007/BF01247837
9 ROSENTHAL, J. S. 1994. Random rotations: characters and random walks on SO N. Ann. Probab. 22 398 423.
Mathematical Reviews (MathSciNet): MR95c:60008
Zentralblatt MATH: 0799.60007
Digital Object Identifier: doi:10.1214/aop/1176988864
Project Euclid: euclid.aop/1176988864
11 SLOANE, N. J. A. 1983. Encry pting by random rotations. Cry ptography. Lecture Notes in Computer Science 149 71 128. Springer, New York.
Mathematical Reviews (MathSciNet): MR85i:94017
Zentralblatt MATH: 0507.94010
12 ZELOBENKO, D. P. 1973. Compact Lie Groups and Their Representations. AMS Trans. Math. Monographs 40. Amer. Math. Soc., Providence, RI.
Mathematical Reviews (MathSciNet): MR57:12776b
BERKELEY, CALIFORNIA 94720 E-MAIL: up@chow.mat.jhu.edu