Electronic Journal of Probability

Mixing time of the fifteen puzzle

Ben Morris and Anastasia Raymer

Full-text: Open access

Abstract

We show that the mixing time for the fifteen puzzle in an $n \times n$ torus is on the order of $n^4 \log n$.

Article information

Source
Electron. J. Probab., Volume 22 (2017), paper no. 9, 29 pp.

Dates
Received: 1 September 2016
Accepted: 17 October 2016
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1485831705

Digital Object Identifier
doi:10.1214/16-EJP11

Mathematical Reviews number (MathSciNet)
MR3613702

Zentralblatt MATH identifier
1360.60136

Subjects
Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Keywords
fifteen puzzle mixing time interchange process

Rights
Creative Commons Attribution 4.0 International License.

Citation

Morris, Ben; Raymer, Anastasia. Mixing time of the fifteen puzzle. Electron. J. Probab. 22 (2017), paper no. 9, 29 pp. doi:10.1214/16-EJP11. https://projecteuclid.org/euclid.ejp/1485831705


Export citation

References

  • [1] Diaconis, P. Group representations in probability and statistics. Institute of Mathematical Statistics, 1988.
  • [2] Diaconis, P. and Saloff-Coste, L. Comparison techniques for random walk on finite groups. Annals of Probability 21 (1993), pp.2131–2156.
  • [3] Diaconis, P. and Saloff-Coste, L. Logarithmic Sobolev inequalities for finite Markov chains. Annals of Applied Probability 6(3) (1996), pp.695–750.
  • [4] Diaconis, P. and Saloff-Coste, L. Random walks on finite groups: a survey of analytic techniques. In Probability Measures on Groups and Related Structures 11 (Z.H. Heyer, ed.) 44–75.
  • [5] Jerrum, M. R. and Sinclair, A. J. (1989). Approximating the permanent. SIAM Journal on Computing 18, 1149–1178.
  • [6] Hoeffding, W. (1963), Probability inequalities for sums of bounded random variables. Journal of the American Statistical Association58 (301), pp. 13–30.
  • [7] Johnson, W. and Story, W. (1879) Notes on the “15” puzzle. American Journal of Mathematics 2 (4), pp.397–404.
  • [8] Lee, T.Y. and Yau, H.T. (1998). Logarithmic Sobolev inequality for some models of random walks. Annals of Probability 26, pp.1855–1873.
  • [9] Levin, D., Peres, Y., and Wilmer, E. Markov chains and mixing times. American Mathematical Society, Providence, RI, 2009.
  • [10] Lezaud, P. Chernoff-type bound for finite Markov chains. Annals of Probability 8, pp.849–867.
  • [11] Morris, B. (2006). The mixing time for simple exclusion. Annals of Applied Probability 16, pp.615–635.
  • [12] Morris, B. and Peres, Y. (2005). Evolving sets, mixing and heat kernel bounds. Probability Theory and Related Fields 133, pp.245–266.
  • [13] Saloff-Coste, L. and Zuniga, J. Refined estimates for some basic random walks on the symmetric and alternating groups, Latin American Journal of Probability and Mathematical Statistics 4, 359-392, 2008.
  • [14] Wilson, D. (2004) Mixing times of lozenge tiling and card shuffling Markov chains. Ann. Appl. Prob. 14, pp. 274–325.
  • [15] Wilson, M. (1974). Graph puzzles, homotopy, and the alternating group. Journal of Combinatorial Theory Series B. 16, pp.86–96.
  • [16] Yau, Horng-Tzer (1997). Logarithmic Sobolev inequality for generalized simple exclusion processes. Probability Theory and Related Fields 109, pp.507–538.