Random walks on the torus with several generators
Open accessAbstract
Given n vectors {i} ∈ [0, 1)d, consider a random walk on the d‐dimensional torus 𝕋d = ℝd/ℤd 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 k th 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/2 d for C2 = C(n, d, j) a constant. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2004
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Integrity signals
Registry
Cites
Statistics
Claims the paper makes
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
The self-check needs the paper’s full text.
Cites
No references could be resolved yet.
Research works printed but not linked: 11
- [1][1] Diaconis, P. Group Representations in Probability and Statistics , Institute of Mathematical Statistics Lecture Notes, Vol. 11, Hayward, CA, 1988. -
- [2][2] Drmota, M, and Tichy, R.F. Sequences, Discrepancies and Applications , Lecture Notes in Math. 1651, 66-71, Springer-Verlag, 1997. -
- [3][3] Gibbs, A. and Su, F.E. On choosing and bounding probability metrics. Internat. Statist. Rev. (2002), vol. 70, no. 3, 419-435. -
- [6][6] Kloss, B.M. Limiting distributions on bicompact topological groups. Theory Probab. Appl. 4 (1959), 237-270. -
- [7][7] Kuipers, L. and Niederreiter, H. Uniform Distribution of Sequences , Wiley, New York, 1974. -
- [8][8] Porod, U. The cut-off phenomenon for random reflections, Ann. Probab. 24 (1996), 74-96. -
- [9][9] Rosenthal, J.S. Random rotations: characters and random walks on SO ( n ), Ann. Probab. 22 (1994), 398-423. -
- [10][10] Schmidt, W.M. Badly approximable systems of linear forms, Journal of Number Theory 1, 139-154, 1969. -
- [11][11] Schmidt, W.M. Diophantine Approximation , Lecture Notes in Math. 785, Springer-Verlag, 1980. -
- [12][12] Su, F.E. Convergence of random walks on the circle generated by an irrational rotation, Trans. Amer. Math. Soc 350 (1998), 3717-3741. -
- [13][13] Su, F.E. A LeVeque type lower bound for discrepancy, Monte Carlo Methods and QuasiMonte Carlo Methods 1998 , 448-458, Springer-Verlag, 2000. -
Cited by
Sorted by citations
Diophantine Gaussian excursions and random walks
Lachièze-Rey, R. (2021) · arXiv · doi:10.48550/arxiv.2104.07290 · Cited by 1 · see citing works
“Then Prescott and Su [28] extended the study in higher dimensional tori.”
Background
- Authors
- Prescott, T., & Su, F. E.
- Affiliations
- University of California, Los Angeles (US)
- Harvey Mudd College (US)
- Corresponding author
- Su, Francis Edward
- Venue
- Random Structures & Algorithms
- Publisher
- Wiley
- Published
- 2004-06-23
- Volume
- 25
- Issue
- 3
- Pages
- 336-345
- ISSN
- 1042-9832 (ISSN-L)
- 1098-2418
- OpenAlex
- W2135570362
- Cited by
- 1
- References
- 0
- Citations by year
- 2022: 1
- FWCI
- 0.00
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Random Structures and Algorithms (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Mathematics › Mathematical Physics › Mathematical Dynamics and Fractals
- Physical Sciences › Mathematics › Numerical Analysis › Mathematical Approximation and Integration
- Physical Sciences › Mathematics › Algebra and Number Theory › Analytic Number Theory Research
- Keywords
- multiple generators, Haar measure, uniform distribution modulo one, step distribution
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
