## The Annals of Applied Probability

### The Glauber dynamics of colorings on trees is rapidly mixing throughout the nonreconstruction regime

#### Abstract

The mixing time of the Glauber dynamics for spin systems on trees is closely related to the reconstruction problem. Martinelli, Sinclair and Weitz established this correspondence for a class of spin systems with soft constraints bounding the log-Sobolev constant by a comparison with the block dynamics [Comm. Math. Phys. 250 (2004) 301–334; Random Structures Algorithms 31 (2007) 134–172]. However, when there are hard constraints, the dynamics inside blocks may be reducible.

We introduce a variant of the block dynamics extending these results to a wide class of spin systems with hard constraints. This applies to essentially any spin system that has nonreconstruction provided that on average the root is not locally frozen in a large neighborhood. In particular, we prove that the mixing time of the Glauber dynamics for colorings on the regular tree is $O(n\log n)$ in the entire nonreconstruction regime.

#### Article information

Source
Ann. Appl. Probab., Volume 27, Number 5 (2017), 2646-2674.

Dates
Revised: July 2016
First available in Project Euclid: 3 November 2017

https://projecteuclid.org/euclid.aoap/1509696031

Digital Object Identifier
doi:10.1214/16-AAP1253

Mathematical Reviews number (MathSciNet)
MR3719943

Zentralblatt MATH identifier
1379.60079

#### Citation

Sly, Allan; Zhang, Yumeng. The Glauber dynamics of colorings on trees is rapidly mixing throughout the nonreconstruction regime. Ann. Appl. Probab. 27 (2017), no. 5, 2646--2674. doi:10.1214/16-AAP1253. https://projecteuclid.org/euclid.aoap/1509696031

#### References

• [1] Berger, N., Kenyon, C., Mossel, E. and Peres, Y. (2005). Glauber dynamics on trees and hyperbolic graphs. Probab. Theory Related Fields 131 311–340.
• [2] Bhatnagar, N., Bhatnagar, N., Vera, J., Vera, J., Vigoda, E., Vigoda, E., Weitz, D. and Weitz, D. (2011). Reconstruction for colorings on trees. SIAM J. Discrete Math. 25 809–826.
• [3] Borgs, C., Chayes, J., Mossel, E. and Roch, S. (2006). The Kesten–Stigum reconstruction bound is tight for roughly symmetric binary channels. In Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science 518–530. IEEE, New York.
• [4] Ding, J., Lubetzky, E. and Peres, Y. (2010). Mixing time of critical Ising model on trees is polynomial in the height. Comm. Math. Phys. 295 161–207.
• [5] Frieze, A. and Vigoda, E. (2007). A survey on the use of Markov chains to randomly sample colourings. In Combinatorics, Complexity, and Chance. Oxford Lecture Ser. Math. Appl. 34 53–71. Oxford Univ. Press, Oxford.
• [6] Georgii, H.-O. (2011). Gibbs Measures and Phase Transitions, 2nd ed. De Gruyter Studies in Mathematics 9. de Gruyter, Berlin.
• [7] Goldberg, L. A., Jerrum, M. and Karpinski, M. (2010). The mixing time of Glauber dynamics for coloring regular trees. Random Structures Algorithms 36 464–476.
• [8] Janson, S. and Mossel, E. (2004). Robust reconstruction on trees is determined by the second eigenvalue. Ann. Probab. 32 2630–2649.
• [9] Jonasson, J. (2002). Uniqueness of uniform random colorings of regular trees. Statist. Probab. Lett. 57 243–248.
• [10] Lucier, B., Molloy, M. and Peres, Y. (2009). The Glauber dynamics for colourings of bounded degree trees. In Approximation, Randomization, and Combinatorial Optimization. Lecture Notes in Computer Science 5687 631–645. Springer, Berlin.
• [11] Martinelli, F. (1999). Lectures on Glauber dynamics for discrete spin models. In Lectures on Probability Theory and Statistics (Saint-Flour, 1997). Lecture Notes in Math. 1717 93–191. Springer, Berlin.
• [12] Martinelli, F., Sinclair, A. and Weitz, D. (2004). Glauber dynamics on trees: Boundary conditions and mixing time. Comm. Math. Phys. 250 301–334.
• [13] Martinelli, F., Sinclair, A. and Weitz, D. (2007). Fast mixing for independent sets, colorings, and other models on trees. Random Structures Algorithms 31 134–172.
• [14] Mossel, E. (2004). Survey: Information flow on trees. In Graphs, Morphisms and Statistical Physics. DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 63 155–170. Amer. Math. Soc., Providence, RI.
• [15] Mossel, E. and Peres, Y. (2003). Information flow on trees. Ann. Appl. Probab. 13 817–844.
• [16] Saloff-Coste, L. (1997). Lectures on finite Markov chains. In Lectures on Probability Theory and Statistics (Saint-Flour, 1996). Lecture Notes in Math. 1665 301–413. Springer, Berlin.
• [17] Sly, A. (2009). Reconstruction of random colourings. Comm. Math. Phys. 288 943–961.
• [18] Sly, A. and Zhang, Y. (2016). The reconstruction threshold for graph coloring is strictly below the rigidity threshold. Preprint. Available at arXiv:1610.02770.
• [19] Tetali, P., Vera, J. C., Vigoda, E. and Yang, L. (2012). Phase transition for the mixing time of the Glauber dynamics for coloring regular trees. Ann. Appl. Probab. 22 2210–2239.
• [20] Vigoda, E. (2000). Improved bounds for sampling colorings. J. Math. Phys. 41 1555–1569.
• [21] Weitz, D. (2004). Mixing in Time and Space for Discrete Spin Systems. Ph.D. thesis, Univ. California, Berkeley. ProQuest LLC, Ann Arbor, MI.