Skip to main content

Fast initial conditions for Glauber dynamics

This article has been updated

Abstract

In the study of Markov chain mixing times, analysis has centered on the performance from a worst-case starting state. Here, in the context of Glauber dynamics for the one-dimensional Ising model, we show how new ideas from information percolation can be used to establish mixing times from other starting states. At high temperatures we show that the alternating initial condition is asymptotically the fastest one, and, surprisingly, its mixing time is faster than at infinite temperature, accelerating as the inverse-temperature \(\beta \) ranges from 0 to \(\beta _0=\frac{1}{2}\mathrm {arctanh}(\frac{1}{3})\). Moreover, the dominant test function depends on the temperature: at \(\beta <\beta _0\) it is autocorrelation, whereas at \(\beta >\beta _0\) it is the Hamiltonian.

This is a preview of subscription content, access via your institution.

Fig. 1

Change history

  • 09 June 2021

    Article note has been updated as Dedicated line in italic

References

  1. Aldous, D.: Random walks on finite groups and rapidly mixing Markov chains. In: Seminar on Probability, vol. XVII, pp. 243–297 (1983)

  2. Aldous, D., Diaconis, P.: Shuffling cards and stopping times. Am. Math. Mon. 93, 333–348 (1986)

    Article  MathSciNet  Google Scholar 

  3. Cox, J.T., Peres, Y., Steif, J.E.: Cutoff for the noisy voter model. Ann. Appl. Probab. 26(2), 917–932 (2016)

    Article  MathSciNet  Google Scholar 

  4. Diaconis, P.: The cutoff phenomenon in finite Markov chains. Proc. Nat. Acad. Sci. USA 93(4), 1659–1664 (1996)

    Article  MathSciNet  Google Scholar 

  5. Diaconis, P., Shahshahani, M.: Generating a random permutation with random transpositions. Z. Wahrsch. Verw. Gebiete 57(2), 159–179 (1981)

    Article  MathSciNet  Google Scholar 

  6. Levin, D.A., Luczak, M., Peres, Y.: Glauber dynamics for the mean-field Ising model: cut-off, critical power law, and metastability. Probab. Theory Relat. Fields 146(1–2), 223–265 (2010)

    Article  MathSciNet  Google Scholar 

  7. Levin, D.A., Peres, Y., Wilmer, E.L.: Markov Chains and Mixing Times. American Mathematical Society, Providence (2008)

    Book  Google Scholar 

  8. Lubetzky, E., Sly, A.: Cutoff for the Ising model on the lattice. Invent. Math. 191(3), 719–755 (2013)

    Article  MathSciNet  Google Scholar 

  9. Lubetzky, E., Sly, A.: Cutoff for general spin systems with arbitrary boundary conditions. Commun. Pure. Appl. Math. 67(6), 982–1027 (2014)

    Article  MathSciNet  Google Scholar 

  10. Lubetzky, E., Sly, A.: An exposition to information percolation for the Ising model. Ann. Fac. Sci. Toulouse Math. (6) 24(4), 745–761 (2015)

    Article  MathSciNet  Google Scholar 

  11. Lubetzky, E., Sly, A.: Information percolation and cutoff for the stochastic Ising model. J. Am. Math. Soc. 29(3), 729–774 (2016)

    Article  MathSciNet  Google Scholar 

  12. Lubetzky, E., Sly, A.: Universality of cutoff for the Ising model. Ann. Probab. 45(6A), 3664–3696 (2017)

    Article  MathSciNet  Google Scholar 

  13. Martinelli, F.: Lectures on Glauber Dynamics for Discrete Spin Models. Lectures on Probability Theory and Statistics (Saint-Flour, 1997). Lecture Notes in Mathematics, vol. 1717, pp. 93–191. Springer, Berlin (1999)

    MATH  Google Scholar 

Download references

Acknowledgements

We thank an anonymous referee for useful comments.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Allan Sly.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Dedicated to the memory of Harry Kesten.

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lubetzky, E., Sly, A. Fast initial conditions for Glauber dynamics. Probab. Theory Relat. Fields 181, 647–667 (2021). https://doi.org/10.1007/s00440-020-01015-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00440-020-01015-3

Mathematics Subject Classification

  • 60J10
  • 60K35