The Annals of Probability

Exact thresholds for Ising–Gibbs samplers on general graphs

Elchanan Mossel and Allan Sly

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Abstract

We establish tight results for rapid mixing of Gibbs samplers for the Ferromagnetic Ising model on general graphs. We show that if

\[(d-1)\tanh\beta<1,\]

then there exists a constant $C$ such that the discrete time mixing time of Gibbs samplers for the ferromagnetic Ising model on any graph of $n$ vertices and maximal degree $d$, where all interactions are bounded by $\beta$, and arbitrary external fields are bounded by $Cn\log n$. Moreover, the spectral gap is uniformly bounded away from $0$ for all such graphs, as well as for infinite graphs of maximal degree $d$.

We further show that when $d\tanh\beta<1$, with high probability over the Erdős–Rényi random graph $G(n,d/n)$, it holds that the mixing time of Gibbs samplers is

\[n^{1+\Theta({1}/{\log\log n})}.\]

Both results are tight, as it is known that the mixing time for random regular and Erdős–Rényi random graphs is, with high probability, exponential in $n$ when $(d-1)\tanh\beta>1$, and $d\tanh\beta>1$, respectively. To our knowledge our results give the first tight sufficient conditions for rapid mixing of spin systems on general graphs. Moreover, our results are the first rigorous results establishing exact thresholds for dynamics on random graphs in terms of spatial thresholds on trees.

Article information

Source
Ann. Probab., Volume 41, Number 1 (2013), 294-328.

Dates
First available in Project Euclid: 23 January 2013

Permanent link to this document
https://projecteuclid.org/euclid.aop/1358951988

Digital Object Identifier
doi:10.1214/11-AOP737

Mathematical Reviews number (MathSciNet)
MR3059200

Zentralblatt MATH identifier
1270.60113

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 82C20: Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs

Keywords
Ising model Glauber dynamics phase transition

Citation

Mossel, Elchanan; Sly, Allan. Exact thresholds for Ising–Gibbs samplers on general graphs. Ann. Probab. 41 (2013), no. 1, 294--328. doi:10.1214/11-AOP737. https://projecteuclid.org/euclid.aop/1358951988


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References

  • [1] Aldous, D. and Fill, J. A. Reversible Markov chains and random walks on graphs. Unpublished manuscript. Available at http://stat-www.berkeley.edu/users/aldous/book.html.
  • [2] Berger, N., Kenyon, C., Mossel, E. and Peres, Y. (2005). Glauber dynamics on prob and hyperbolic graphs. Probab. Theory Related Fields 131 311–340.
  • [3] Cesi, F. (2001). Quasi-factorization of the entropy and logarithmic Sobolev inequalities for Gibbs random fields. Probab. Theory Related Fields 120 569–584.
  • [4] Dembo, A. and Montanari, A. (2010). Ising models on locally tree-like graphs. Ann. Appl. Probab. 20 565–592.
  • [5] Dobrushin, R. L. and Shlosman, S. B. (1985). Constructive criterion for uniqueness of a Gibbs field. In Statistical Mechanics and Dynamical Systems, Volume 10 (J. Fritz, A. Jaffe and D. Szasz, eds.) 347–370.
  • [6] Dyer, M., Sinclair, A., Vigoda, E. and Weitz, D. (2004). Mixing in time and space for lattice spin systems: A combinatorial view. Random Structures Algorithms 24 461–479.
  • [7] Gershchenfeld, A. and Montanari, A. (2007). Reconstruction for models on random graphs. In Annual IEEE Symposium on Foundations of Computer Science 194–204. IEEE Comput. Soc., Los Alamitos, CA.
  • [8] Hayes, T. P. and Sinclair, A. (2007). A general lower bound for mixing of single-site dynamics on graphs. Ann. Appl. Probab. 17 931–952.
  • [9] Hayes, T. P. and Vigoda, E. (2006). Coupling with the stationary distribution and improved sampling for colorings and independent sets. Ann. Appl. Probab. 16 1297–1318.
  • [10] Higuchi, Y. (1993). Coexistence of infinite (∗)-clusters II. Ising percolation in two dimensions. Probab. Theory Related Fields 97 1–33.
  • [11] Jerrum, M. and Sinclair, A. (1989). Approximating the permanent. SIAM J. Comput. 18 1149–1178.
  • [12] Kenyon, C., Mossel, E. and Peres, Y. (2001). Glauber dynamics on trees and hyperbolic graphs. In 42nd IEEE Symposium on Foundations of Computer Science (Las Vegas, NV) 568–578. IEEE Comput. Soc., Los Alamitos, CA.
  • [13] Krz̧akała, F., Montanari, A., Ricci-Tersenghi, F., Semerjian, G. and Zdeborová, L. (2007). Gibbs states and the set of solutions of random constraint satisfaction problems. Proc. Natl. Acad. Sci. USA 104 10318–10323.
  • [14] Levin, D. A., Peres, Y. and Wilmer, E. L. (2009). Markov Chains and Mixing Times. Amer. Math. Soc., Providence, RI. With a chapter by James G. Propp and David B. Wilson.
  • [15] Liggett, T. M. (1985). Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 276. Springer, New York.
  • [16] Lyons, R. (1989). The Ising model and percolation on trees and tree-like graphs. Comm. Math. Phys. 125 337–353.
  • [17] Martinelli, F. (1999). Lectures on Glauber dynamics for discrete spin models. In Lectures on Probability Theory and Statistics. Lecture Notes in Math. 1717 93–191. Springer, Berlin.
  • [18] Martinelli, F. and Olivieri, E. (1994). Approach to equilibrium of Glauber dynamics in the one phase region. I. The attractive case. Comm. Math. Phys. 161 447–486.
  • [19] Martinelli, F. and Olivieri, E. (1994). Approach to equilibrium of Glauber dynamics in the one phase region. II. The general case. Comm. Math. Phys. 161 487–514.
  • [20] Martinelli, F., Sinclair, A. and Weitz, D. (2003). The Ising model on trees: Boundary conditions and mixing time. In Proceedings of the Forty Fourth Annual Symposium on Foundations of Computer Science 628–639.
  • [21] Martinelli, F., Sinclair, A. A. and Weitz, D. (2004). Glauber dynamics on trees: Boundary conditions and mixing time. Comm. Math. Phys. 250 301–334.
  • [22] Mézard, M. and Montanari, A. (2009). Information, Physics, and Computation. Oxford Univ. Press, Oxford.
  • [23] Montanari, A., Ricci-Tersenghi, F. and Semerjian, G. (2008). Clusters of solutions and replica symmetry breaking in random k-satisfiability. J. Stat. Mech. Theory Exp. 2008 P04004.
  • [24] Mossel, E. and Sly, A. (2008). Rapid mixing of Gibbs sampling on graphs that are sparse on average. In Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA) 238–247.
  • [25] Mossel, E. and Sly, A. (2009). Rapid mixing of Gibbs sampling on graphs that are sparse on average. Random Structures Algorithms 35 250–270.
  • [26] Mossel, E., Weitz, D. and Wormald, N. (2009). On the hardness of sampling independent sets beyond the tree threshold. Probab. Theory Related Fields 143 401–439.
  • [27] Peres, Y. Mixing for Markov chains and spin systems. Draft lecture notes.
  • [28] Stroock, D. W. and Zegarliński, B. (1992). The logarithmic Sobolev inequality for discrete spin systems on a lattice. Comm. Math. Phys. 149 175–193.
  • [29] Vigoda, E. (1999). Improved bounds for sampling coloring. In 40th Annual Symposium on Foundations of Computer Science (FOCS) 51–59.
  • [30] Vigoda, E. (2000). Improved bounds for sampling coloring. J. Math. Phys. 3 1555–1569.
  • [31] Weitz, D. (2005). Combinatorial criteria for uniqueness of Gibbs measures. Random Structures Algorithms 27 445–475.
  • [32] Weitz, D. (2006). Counting indpendent sets up to the tree threshold. In Proceedings of the Thirty-eighth Annual ACM Symposium on Theory of Computing 140–149. ACM, New York.
  • [33] Zhang, J., Liang, H. and Bai, F. (2011). Approximating partition functions of the two-state spin system. Inform. Process. Lett. 111 702–710.