No CrossRef data available.
Article contents
The hit-and-run version of top-to-random
Part of:
Markov processes
Probability theory on algebraic and topological structures
Stochastic processes
Published online by Cambridge University Press: 12 July 2022
Abstract
We study an example of a hit-and-run random walk on the symmetric group 
$\mathbf S_n$
. Our starting point is the well-understood top-to-random shuffle. In the hit-and-run version, at each single step, after picking the point of insertion j uniformly at random in 
$\{1,\ldots,n\}$
, the top card is inserted in the jth position k times in a row, where k is uniform in 
$\{0,1,\ldots,j-1\}$
. The question is, does this accelerate mixing significantly or not? We show that, in 
$L^2$
and sup-norm, this accelerates mixing at most by a constant factor (independent of n). Analyzing this problem in total variation is an interesting open question. We show that, in general, hit-and-run random walks on finite groups have non-negative spectrum.
- Type
- Original Article
- Information
- Copyright
- © The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust
References
Aldous, A. (1983). Random walks on finite groups and rapidly mixing Markov chains. In Seminar on Probability XVII (Lecture Notes Math. 986), pp. 243–297. Springer, Berlin.10.1007/BFb0068322CrossRefGoogle Scholar
Bernstein, M. and Nestoridi, E. (2019). Cutoff for random to random card shuffle. Ann. Prob. 47, 3303–3320.10.1214/19-AOP1340CrossRefGoogle Scholar
Brown, K. S. and Diaconis, P. (1998). Random walks and hyperplane arrangements. Ann. Prob. 26, 1813–1854.10.1214/aop/1022855884CrossRefGoogle Scholar
Chen, G.-Y. and Saloff-Coste, L. (2008). The cutoff phenomenon for ergodic Markov processes. Electron. J. Prob. 13, 26–78.10.1214/EJP.v13-474CrossRefGoogle Scholar
Diaconis, P. (1988). Group Representations in Probability and Statistics (Institute of Mathematical Statistics Lecture Notes: Monograph Series 11). Institute of Mathematical Statistics, Hayward, CA.Google Scholar
Diaconis, P. (1996). The cutoff phenomenon in finite Markov chains. Proc. Nat. Acad. Sci. USA 93, 1659–1664.10.1073/pnas.93.4.1659CrossRefGoogle Scholar
Diaconis, P. and Saloff-Coste, L. (1993). Comparison techniques for random walk on finite groups. Ann. Prob. 21, 2131–2156.10.1214/aop/1176989013CrossRefGoogle Scholar
Diaconis, P. and Shahshahani, M. (1981). Generating a random permutation with random transpositions. Z. Wahrscheinlichkeitsth. 57, 159–179.10.1007/BF00535487CrossRefGoogle Scholar
Diaconis, P., Fill, J. A. and Pitman, J. (1992). Analysis of top to random shuffles. Combinatorics Prob. Comput. 1, 135–155.10.1017/S0963548300000158CrossRefGoogle Scholar
Dyer, M., Greenhill, C. and Ullrich, M. (2014). Structure and eigenvalues of heat-bath Markov chains. Linear Algebra Appl. 454, 57–71.10.1016/j.laa.2014.04.018CrossRefGoogle Scholar
Rudolf, D. and Ullrich, M. (2013). Positivity of hit-and-run and related algorithms. Electron. Commun. Prob. 18, 8.10.1214/ECP.v18-2507CrossRefGoogle Scholar
Saloff-Coste, L. (1997). Lectures on finite Markov chains. In Lectures on Probability Theory and Statistics (Lecture Notes Math. 1665), ed. P. Bernard, pp. 301–413. Springer, Berlin.10.1007/BFb0092621CrossRefGoogle Scholar
Saloff-Coste, L. (2004). Random walks on finite groups. In Probability on Discrete Structures (Encyclopaedia Math. Sci. 110), pp. 263–346. Springer, Berlin.10.1007/978-3-662-09444-0_5CrossRefGoogle Scholar
Saloff-Coste, L. and Zúñiga, J. (2008). Refined estimates for some basic random walks on the symmetric and alternating groups. ALEA Lat. Am. J. Prob. Math. Statist. 4, 359–392.Google Scholar
Ullrich, M. (2014). Rapid mixing of Swendsen–Wang dynamics in two dimensions. Dissertationes Math. 502, 1–64.10.4064/dm502-0-1CrossRefGoogle Scholar