Annals of Probability

Finitary codings for spatial mixing Markov random fields

Yinon Spinka

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

It has been shown by van den Berg and Steif (Ann. Probab. 27 (1999) 1501–1522) that the subcritical and critical Ising model on $\mathbb{Z}^{d}$ is a finitary factor of an i.i.d. process (ffiid), whereas the super-critical model is not. In fact, they showed that the latter is a general phenomenon in that a phase transition presents an obstruction for being ffiid. The question remained whether this is the only such obstruction. We make progress on this, showing that certain spatial mixing conditions (notions of weak dependence on boundary conditions, not to be confused with other notions of mixing in ergodic theory) imply ffiid. Our main result is that weak spatial mixing implies ffiid with power-law tails for the coding radius, and that strong spatial mixing implies ffiid with exponential tails for the coding radius. The weak spatial mixing condition can be relaxed to a condition which is satisfied by some critical two-dimensional models. Using a result of the author (Spinka (2018)), we deduce that strong spatial mixing also implies ffiid with stretched-exponential tails from a finite-valued i.i.d. process.

We give several applications to models such as the Potts model, proper colorings, the hard-core model, the Widom–Rowlinson model and the beach model. For instance, for the ferromagnetic $q$-state Potts model on $\mathbb{Z}^{d}$ at inverse temperature $\beta $, we show that it is ffiid with exponential tails if $\beta $ is sufficiently small, it is ffiid if $\beta <\beta _{c}(q,d)$, it is not ffiid if $\beta >\beta_{c}(q,d)$ and, when $d=2$ and $\beta =\beta _{c}(q,d)$, it is ffiid if and only if $q\le 4$.

Article information

Source
Ann. Probab., Volume 48, Number 3 (2020), 1557-1591.

Dates
Received: May 2018
First available in Project Euclid: 17 June 2020

Permanent link to this document
https://projecteuclid.org/euclid.aop/1592359238

Digital Object Identifier
doi:10.1214/19-AOP1405

Mathematical Reviews number (MathSciNet)
MR4112724

Subjects
Primary: 60J99: None of the above, but in this section 60G10: Stationary processes
Secondary: 37A60: Dynamical systems in statistical mechanics [See also 82Cxx] 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 28D99: None of the above, but in this section

Keywords
Finitary factor Markov random field spatial mixing

Citation

Spinka, Yinon. Finitary codings for spatial mixing Markov random fields. Ann. Probab. 48 (2020), no. 3, 1557--1591. doi:10.1214/19-AOP1405. https://projecteuclid.org/euclid.aop/1592359238


Export citation

References

  • [1] Achlioptas, D., Molloy, M., Moore, C. and Van Bussel, F. (2005). Rapid mixing for lattice colourings with fewer colours. J. Stat. Mech. Theory Exp. 2005 P10012.
  • [2] Aizenman, M., Duminil-Copin, H. and Sidoravicius, V. (2015). Random currents and continuity of Ising model’s spontaneous magnetization. Comm. Math. Phys. 334 719–742.
  • [3] Aizenman, M. and Fernández, R. (1986). On the critical behavior of the magnetization in high-dimensional Ising models. J. Stat. Phys. 44 393–454.
  • [4] Akcoglu, M. A., del Junco, A. and Rahe, M. (1979). Finitary codes between Markov processes. Z. Wahrsch. Verw. Gebiete 47 305–314.
  • [5] Alexander, K. S. (1998). On weak mixing in lattice models. Probab. Theory Related Fields 110 441–471.
  • [6] Alexander, K. S. (2004). Mixing properties and exponential decay for lattice systems in finite volumes. Ann. Probab. 32 441–487.
  • [7] Angel, O., Benjamini, I., Gurel-Gurevich, O., Meyerovitch, T. and Peled, R. (2012). Stationary map coloring. Ann. Inst. Henri Poincaré Probab. Stat. 48 327–342.
  • [8] Angel, O. and Spinka, Y. (2019). Pairwise optimal coupling of multiple random variables. Preprint. Available at arXiv:1903.00632.
  • [9] Blanca, A., Caputo, P., Sinclair, A. and Vigoda, E. (2018). Spatial mixing and non-local Markov chains. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms 1965–1980. SIAM, Philadelphia, PA.
  • [10] Bosco, G. G., Machado, F. P. and Ritchie, T. L. (2010). Exponential rates of convergence in the ergodic theorem: A constructive approach. J. Stat. Phys. 139 367–374.
  • [11] Briceño, R. (2018). The topological strong spatial mixing property and new conditions for pressure approximation. Ergodic Theory Dynam. Systems 38 1658–1696.
  • [12] Briceño, R. and Pavlov, R. (2017). Strong spatial mixing in homomorphism spaces. SIAM J. Discrete Math. 31 2110–2137.
  • [13] Brightwell, G. R., Häggström, O. and Winkler, P. (1999). Nonmonotonic behavior in hard-core and Widom–Rowlinson models. J. Stat. Phys. 94 415–435.
  • [14] Brightwell, G. R. and Winkler, P. (2000). Gibbs measures and dismantlable graphs. J. Combin. Theory Ser. B 78 141–166.
  • [15] Burton, R. and Steif, J. E. (1994). Non-uniqueness of measures of maximal entropy for subshifts of finite type. Ergodic Theory Dynam. Systems 14 213–235.
  • [16] Burton, R. and Steif, J. E. (1995). New results on measures of maximal entropy. Israel J. Math. 89 275–300.
  • [17] Bušic, A., Mairesse, J. and Marcovici, I. (2013). Probabilistic cellular automata, invariant measures, and perfect sampling. Adv. in Appl. Probab. 45 960–980.
  • [18] De Santis, E. and Lissandrelli, A. (2012). Developments in perfect simulation of Gibbs measures through a new result for the extinction of Galton–Watson-like processes. J. Stat. Phys. 147 231–251.
  • [19] De Santis, E. and Piccioni, M. (2008). Exact simulation for discrete time spin systems and unilateral fields. Methodol. Comput. Appl. Probab. 10 105–120.
  • [20] del Junco, A. (1980). Finitary coding of Markov random fields. Z. Wahrsch. Verw. Gebiete 52 193–202.
  • [21] del Junco, A. (1990). Bernoulli shifts of the same entropy are finitarily and unilaterally isomorphic. Ergodic Theory Dynam. Systems 10 687–715.
  • [22] den Hollander, F. and Steif, J. E. (2006). Random walk in random scenery: A survey of some recent results. In Dynamics & Stochastics. Institute of Mathematical Statistics Lecture Notes—Monograph Series 48 53–65. IMS, Beachwood, OH.
  • [23] Diaconis, P. and Freedman, D. (1999). Iterated random functions. SIAM Rev. 41 45–76.
  • [24] Dimakos, X. K. (2001). A guide to exact simulation. Int. Stat. Rev. 69 27–48.
  • [25] Dobrushin, R. L. (1968). The description of a random field by means of conditional probabilities and conditions of its regularity. Theory Probab. Appl. 13 197–224.
  • [26] Dobrushin, R. L. and Shlosman, S. B. (1985). Completely analytical Gibbs fields. In Statistical Physics and Dynamical Systems (Köszeg, 1984). Progress in Probability 10 371–403. Birkhäuser, Boston, MA.
  • [27] Dobrushin, R. L. and Shlosman, S. B. (1985). Constructive criterion for the uniqueness of Gibbs field. In Statistical Physics and Dynamical Systems (Köszeg, 1984). Progress in Probability 10 347–370. Birkhäuser, Boston, MA.
  • [28] Duminil-Copin, H., Gagnebin, M., Harel, M., Manolescu, I. and Tassion, V. (2016). Discontinuity of the phase transition for the planar random-cluster and Potts models with ${q>4}$. Preprint. Available at arXiv:1611.09877.
  • [29] Duminil-Copin, H., Raoufi, A. and Tassion, V. (2019). Sharp phase transition for the random-cluster and Potts models via decision trees. Ann. of Math. (2) 189 75–99.
  • [30] Duminil-Copin, H., Sidoravicius, V. and Tassion, V. (2017). Continuity of the phase transition for planar random-cluster and Potts models with ${1\leq q\leq 4}$. Comm. Math. Phys. 349 47–107.
  • [31] 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.
  • [32] Edwards, R. G. and Sokal, A. D. (1988). Generalization of the Fortuin–Kasteleyn–Swendsen–Wang representation and Monte Carlo algorithm. Phys. Rev. D 38 2009–2012.
  • [33] Fiebig, U.-R. (1984). A return time invariant for finitary isomorphisms. Ergodic Theory Dynam. Systems 4 225–231.
  • [34] Fill, J. A. and Machida, M. (2001). Stochastic monotonicity and realizable monotonicity. Ann. Probab. 29 938–978.
  • [35] Galves, A., Garcia, N. and Löcherbach, E. (2008). Perfect simulation and finitary coding for multicolor systems with interactions of infinite range. Preprint. Available at arXiv:0809.3494.
  • [36] Galves, A., Löcherbach, E. and Orlandi, E. (2010). Perfect simulation of infinite range Gibbs measures and coupling with their finite range approximations. J. Stat. Phys. 138 476–495.
  • [37] Gamarnik, D., Katz, D. and Misra, S. (2015). Strong spatial mixing of list coloring of graphs. Random Structures Algorithms 46 599–613.
  • [38] Georgii, H.-O., Häggström, O. and Maes, C. (2001). The random geometry of equilibrium phases. In Phase Transitions and Critical Phenomena, Vol. 18. Phase Transit. Crit. Phenom. 18 1–142. Academic Press, San Diego, CA.
  • [39] Goldberg, L. A., Jalsenius, M., Martin, R. and Paterson, M. (2006). Improved mixing bounds for the anti-ferromagnetic Potts model on ${\Bbb{Z}}^{2}$. LMS J. Comput. Math. 9 1–20.
  • [40] Goldberg, L. A., Martin, R. and Paterson, M. (2005). Strong spatial mixing with fewer colors for lattice graphs. SIAM J. Comput. 35 486–517.
  • [41] Grimmett, G. (2006). The Random-Cluster Model. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 333. Springer, Berlin.
  • [42] Häggström, O. (1996). On phase transitions for subshifts of finite type. Israel J. Math. 94 319–352.
  • [43] Häggström, O. (2002). A monotonicity result for hard-core and Widom–Rowlinson models on certain $d$-dimensional lattices. Electron. Commun. Probab. 7 67–78.
  • [44] Häggström, O. and Nelander, K. (1998). Exact sampling from anti-monotone systems. Stat. Neerl. 52 360–380.
  • [45] Häggström, O. and Nelander, K. (1999). On exact simulation of Markov random fields using coupling from the past. Scand. J. Stat. 26 395–411.
  • [46] Häggström, O. and Steif, J. E. (2000). Propp–Wilson algorithms and finitary codings for high noise Markov random fields. Combin. Probab. Comput. 9 425–439.
  • [47] Harel, M. and Spinka, Y. (2018). Finitary codings for the random-cluster model and other infinite-range monotone models. Preprint. Available at arXiv:1808.02333.
  • [48] Harvey, N., Holroyd, A. E., Peres, Y. and Romik, D. (2007). Universal finitary codes with exponential tails. Proc. Lond. Math. Soc. (3) 94 475–496.
  • [49] Harvey, N. and Peres, Y. (2011). An invariant of finitary codes with finite expected square root coding length. Ergodic Theory Dynam. Systems 31 77–90.
  • [50] Holroyd, A. E. (2017). One-dependent coloring by finitary factors. Ann. Inst. Henri Poincaré Probab. Stat. 53 753–765.
  • [51] Holroyd, A. E., Hutchcroft, T. and Levy, A. (2017). Mallows permutations and finite dependence. Preprint. Available at arXiv:1706.09526.
  • [52] Holroyd, A. E., Schramm, O. and Wilson, D. B. (2017). Finitary coloring. Ann. Probab. 45 2867–2898.
  • [53] Huber, M. (1998). Exact sampling and approximate counting techniques. In Proceedings of the Thirtieth Annual ACM Symposium on Theory of Computing 31–40. ACM, New York.
  • [54] Huber, M. (2004). Perfect sampling using bounding chains. Ann. Appl. Probab. 14 734–753.
  • [55] Jalsenius, M. (2009). Strong spatial mixing and rapid mixing with five colours for the Kagome lattice. LMS J. Comput. Math. 12 195–227.
  • [56] Katznelson, Y. and Weiss, B. (1972). Commuting measure-preserving transformations. Israel J. Math. 12 161–173.
  • [57] Keane, M. and Smorodinsky, M. (1977). A class of finitary codes. Israel J. Math. 26 352–371.
  • [58] Keane, M. and Smorodinsky, M. (1979). Bernoulli schemes of the same entropy are finitarily isomorphic. Ann. of Math. (2) 109 397–406.
  • [59] Keane, M. and Smorodinsky, M. (1979). Finitary isomorphisms of irreducible Markov shifts. Israel J. Math. 34 281–286.
  • [60] Keane, M. and Steif, J. E. (2003). Finitary coding for the one-dimensional $T,T^{-1}$ process with drift. Ann. Probab. 31 1979–1985.
  • [61] Kendall, W. S. and Møller, J. (2000). Perfect simulation using dominating processes on ordered spaces, with application to locally stable point processes. Adv. in Appl. Probab. 32 844–865.
  • [62] Krieger, W. (1983). On the finitary isomorphisms of Markov shifts that have finite expected coding time. Z. Wahrsch. Verw. Gebiete 65 323–328.
  • [63] Lebowitz, J. and Gallavotti, G. (1971). Phase transitions in binary lattice gases. J. Math. Phys. 12 1129–1133.
  • [64] Li, L., Lu, P. and Yin, Y. (2013). Correlation decay up to uniqueness in spin systems. In Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms 67–84. SIAM, Philadelphia.
  • [65] Lubetzky, E. and Sly, A. (2012). Critical Ising on the square lattice mixes in polynomial time. Comm. Math. Phys. 313 815–836.
  • [66] Lubetzky, E. and Sly, A. (2013). Cutoff for the Ising model on the lattice. Invent. Math. 191 719–755.
  • [67] Lubetzky, E. and Sly, A. (2014). Cutoff for general spin systems with arbitrary boundary conditions. Comm. Pure Appl. Math. 67 982–1027.
  • [68] Lyons, R. and Nazarov, F. (2011). Perfect matchings as IID factors on non-amenable groups. European J. Combin. 32 1115–1125.
  • [69] Marcus, B. and Pavlov, R. (2013). Computing bounds for entropy of stationary $\Bbb{Z}^{d}$ Markov random fields. SIAM J. Discrete Math. 27 1544–1558.
  • [70] 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.
  • [71] 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.
  • [72] 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.
  • [73] Martinelli, F., Olivieri, E. and Schonmann, R. H. (1994). For $2$-D lattice spin systems weak mixing implies strong mixing. Comm. Math. Phys. 165 33–47.
  • [74] Marton, K. and Shields, P. C. (1994). The positive-divergence and blowing-up properties. Israel J. Math. 86 331–348.
  • [75] Mešalkin, L. D. (1959). A case of isomorphism of Bernoulli schemes. Dokl. Akad. Nauk SSSR 128 41–44.
  • [76] 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 238–247. ACM, New York.
  • [77] Nair, C. and Tetali, P. (2007). The correlation decay (CD) tree and strong spatial mixing in multi-spin systems. Preprint. Available at arXiv:math/0701494.
  • [78] Ornstein, D. (1970). Bernoulli shifts with the same entropy are isomorphic. Adv. Math. 4 337–352.
  • [79] Peled, R. and Samotij, W. (2014). Odd cutsets and the hard-core model on $\Bbb{Z}^{d}$. Ann. Inst. Henri Poincaré Probab. Stat. 50 975–998.
  • [80] Peled, R. and Spinka, Y. (2017). A condition for long-range order in discrete spin systems with application to the antiferromagnetic Potts model. Preprint. Available at arXiv:1712.03699.
  • [81] Peled, R. and Spinka, Y. (2018). Rigidity of proper colorings of ${\mathbb{Z}}^{d}$. Preprint. Available at arXiv:1808.03597.
  • [82] Propp, J. and Wilson, D. (1998). Coupling from the past: A user’s guide. In Microsurveys in Discrete Probability (Princeton, NJ, 1997). DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 41 181–192. Amer. Math. Soc., Providence, RI.
  • [83] Propp, J. G. and Wilson, D. B. (1996). Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Structures Algorithms 9 223–252.
  • [84] Ray, G. and Spinka, Y. (2019). A short proof of the discontinuity of phase transition in the planar random-cluster model with ${q>4}$. Preprint. Available at arXiv:1904.10557.
  • [85] Rudolph, D. J. (1981). A characterization of those processes finitarily isomorphic to a Bernoulli shift. In Ergodic Theory and Dynamical Systems, I (College Park, MD., 1979–80). Progr. Math. 10 1–64. Birkhäuser, Boston, MA.
  • [86] Rudolph, D. J. (1982). A mixing Markov chain with exponentially decaying return times is finitarily Bernoulli. Ergodic Theory Dynam. Systems 2 85–97.
  • [87] Runnels, L. and Lebowitz, J. (1974). Phase transitions of a multicomponent Widom–Rowlinson model. J. Math. Phys. 15 1712–1717.
  • [88] Salas, J. and Sokal, A. D. (1997). Absence of phase transition for antiferromagnetic Potts models via the Dobrushin uniqueness theorem. J. Stat. Phys. 86 551–579.
  • [89] Schmidt, K. (1984). Invariants for finitary isomorphisms with finite expected code lengths. Invent. Math. 76 33–40.
  • [90] Serafin, J. (2006). Finitary codes, a short survey. In Dynamics & Stochastics. Institute of Mathematical Statistics Lecture Notes—Monograph Series 48 262–273. IMS, Beachwood, OH.
  • [91] Sinclair, A., Srivastava, P., Štefankovic, D. and Yin, Y. (2017). Spatial mixing and the connective constant: Optimal bounds. Probab. Theory Related Fields 168 153–197.
  • [92] Slawny, J. (1981). Ergodic properties of equilibrium states. Comm. Math. Phys. 80 477–483.
  • [93] Smorodinsky, M. (1992). Finitary isomorphism of $m$-dependent processes. In Symbolic Dynamics and Its Applications (New Haven, CT, 1991). Contemp. Math. 135 373–376. Amer. Math. Soc., Providence, RI.
  • [94] Soo, T. (2010). Translation-equivariant matchings of coin flips on $\Bbb{Z}^{d}$. Adv. in Appl. Probab. 42 69–82.
  • [95] Spinka, Y. (2018). Finitary coding for the sub-critical Ising model with finite expected coding volume. Preprint. Available at arXiv:1801.02529.
  • [96] Steif, J. E. (2001). The $T,T^{-1}$ process, finitary codings and weak Bernoulli. Israel J. Math. 125 29–43.
  • [97] Strassen, V. (1965). The existence of probability measures with given marginals. Ann. Math. Stat. 36 423–439.
  • [98] Stroock, D. W. and Zegarlinski, B. (1992). The logarithmic Sobolev inequality for discrete spin systems on a lattice. Comm. Math. Phys. 149 175–193.
  • [99] Swendsen, R. H. and Wang, J.-S. (1987). Nonuniversal critical dynamics in Monte Carlo simulations. Phys. Rev. Lett. 58 86.
  • [100] Timar, A. (2009). Invariant matchings of exponential tail on coin flips in $\mathbf{Z}^{d}$. Preprint. Available at arXiv:0909.1090.
  • [101] Timár, Á. (2011). Invariant colorings of random planar maps. Ergodic Theory Dynam. Systems 31 549–562.
  • [102] van den Berg, J. (1999). On the absence of phase transition in the monomer-dimer model. In Perplexing Problems in Probability. Progress in Probability 44 185–195. Birkhäuser, Boston, MA.
  • [103] van den Berg, J. and Maes, C. (1994). Disagreement percolation in the study of Markov fields. Ann. Probab. 22 749–763.
  • [104] van den Berg, J. and Steif, J. E. (1994). Percolation and the hard-core lattice gas model. Stochastic Process. Appl. 49 179–197.
  • [105] van den Berg, J. and Steif, J. E. (1999). On the existence and nonexistence of finitary codings for a class of random fields. Ann. Probab. 27 1501–1522.
  • [106] Weitz, D. (2004). Mixing in Time and Space for Discrete Spin Systems. ProQuest LLC, Ann Arbor, MI. Ph.D. thesis, University of California, Berkeley.
  • [107] Weitz, D. (2005). Combinatorial criteria for uniqueness of Gibbs measures. Random Structures Algorithms 27 445–475.
  • [108] Weitz, D. (2006). Counting independent sets up to the tree threshold. In STOC’06: Proceedings of the 38th Annual ACM Symposium on Theory of Computing 140–149. ACM, New York.
  • [109] Yang, C. N. (1952). The spontaneous magnetization of a two-dimensional Ising model. Phys. Rev. 85 808–816.
  • [110] Yin, Y. and Zhang, C. (2015). Spatial mixing and approximate counting for Potts model on graphs with bounded average degree. Preprint. Available at arXiv:1507.07225.