Random Walks on the Torus with Several Generators

Authors

Timothy Prescott '02, Harvey Mudd CollegeFollow
Francis E. Su, Harvey Mudd CollegeFollow

Student Co-author

HMC Undergraduate

Document Type

Article - preprint

Department

Mathematics (HMC)

Publication Date

10-2004

Abstract

Given n vectors {_i} ∈ [0, 1)^d, consider a random walk on the d-dimensional torus ^d = ℝ^d/ℤ^d generated by these vectors by successive addition and subtraction. For certain sets of vectors, this walk converges to Haar (uniform) measure on the torus. We show that the discrepancy distance D(Q*^k) between the kth step distribution of the walk and Haar measure is bounded below by D(Q*^k) ≥ C₁k^−n/2, where C₁ = C(n, d) is a constant. If the vectors are badly approximated by rationals (in a sense we will define), then D(Q*^k) ≤ C₂k^−n/2d for C₂ = C(n, d, _j) a constant.

Comments

Author's pre-print manuscript available for download.

The definitive version is available at http://dx.doi.org/10.1002/rsa.20029.

Rights Information

© 2004 Wiley Periodicals

Terms of Use & License Information

Terms of Use for work posted in Scholarship@Claremont.

DOI

10.1002/rsa.20029

Recommended Citation

Timothy Prescott and Francis Edward Su. Random walks on the torus with several generators. Random Structures and Algorithms, 25(3):336–345, 2004.