Book contents
- Frontmatter
- Contents
- Preface
- Part I Finite Abelian Groups
- 1 Congruences and the Quotient Ring of the Integers mod n
- 2 The Discrete Fourier Transform on the Finite Circle ℤ/nℤ
- 3 Graphs of ℤ/nℤ, Adjacency Operators, Eigenvalues
- 4 Four Questions about Cayley Graphs
- 5 Finite Euclidean Graphs and Three Questions about Their Spectra
- 6 Random Walks on Cayley Graphs
- 7 Applications in Geometry and Analysis. Connections between Continuous and Finite Problems. Dido's Problem for Polygons
- 8 The Quadratic Reciprocity Law
- 9 The Fast Fourier Transform or FFT
- 10 The DFT on Finite Abelian Groups – Finite Tori
- 11 Error-Correcting Codes
- 12 The Poisson Sum Formula on a Finite Abelian Group
- 13 Some Applications in Chemistry and Physics
- 14 The Uncertainty Principle
- Part II Finite Nonabelian Groups
- References
- Index
6 - Random Walks on Cayley Graphs
from Part I - Finite Abelian Groups
Published online by Cambridge University Press: 06 July 2010
- Frontmatter
- Contents
- Preface
- Part I Finite Abelian Groups
- 1 Congruences and the Quotient Ring of the Integers mod n
- 2 The Discrete Fourier Transform on the Finite Circle ℤ/nℤ
- 3 Graphs of ℤ/nℤ, Adjacency Operators, Eigenvalues
- 4 Four Questions about Cayley Graphs
- 5 Finite Euclidean Graphs and Three Questions about Their Spectra
- 6 Random Walks on Cayley Graphs
- 7 Applications in Geometry and Analysis. Connections between Continuous and Finite Problems. Dido's Problem for Polygons
- 8 The Quadratic Reciprocity Law
- 9 The Fast Fourier Transform or FFT
- 10 The DFT on Finite Abelian Groups – Finite Tori
- 11 Error-Correcting Codes
- 12 The Poisson Sum Formula on a Finite Abelian Group
- 13 Some Applications in Chemistry and Physics
- 14 The Uncertainty Principle
- Part II Finite Nonabelian Groups
- References
- Index
Summary
Anyone who considers arithmetical methods of producing random digits is, of course, in a state of sin.
J. von NeumannIn the first part of this chapter we obtain limit theorems for the simplest random walks on ℤ/nℤ, for n odd, using the DFT and Markov chains. In the second part we redo some of the first part, replacing Markov chains with sums of random variables. We begin with the example of random number generators.
References for this chapter include: Fan Chung [1996], F. Chung, P. Diaconis, and R. Graham [1987], P. Diaconis [1988], P. Diaconis and M. Shashahani [1986], P. Diaconis and D. Stroock [1987], P. Doyle and J. L. Snell [1984], W. Feller [1968], R. Guy [1984, Vol. 3, Section K45], J. G. Kemeny and J. L. Snell [1960], W. LeVeque [1974, Vol. 3, Section K45], K. Rosen [1993, Section 8.7], J. T. Sandefur [1990], J. L. Snell [1975], and M. Schroeder [1986, Chapter 27]. See also Chapters 17 and 18 for more information on random number generators.
Random Number Generators
There are many reasons why programs such as Mathematica and Matlab are capable of giving us a random number at the drop of a hat or the push of a key. Applications include computer simulations, sampling, testing of computer algorithms, decision making, Monte Carlo methods for numerical integration, and fault detection.
In the “good” old days, people obtained random numbers from tables.
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- Information
- Fourier Analysis on Finite Groups and Applications , pp. 97 - 113Publisher: Cambridge University PressPrint publication year: 1999