## Mixing in time and space for discrete spin systems (2004)

Citations: | 11 - 3 self |

### BibTeX

@TECHREPORT{Weitz04mixingin,

author = {Dror Weitz},

title = {Mixing in time and space for discrete spin systems},

institution = {},

year = {2004}

}

### OpenURL

### Abstract

### Citations

1563 |
Exactly Solved Models in Statistical Mechanics
- Baxter
- 1982
(Show Context)
Citation Context ...·). (If z is the root of Tb then p(·) = µ τ Bz,ℓ (·).) p(σz=+) as σ We now describe a recursive calculation of the magnetization Rz,ℓ, the details of which (up to change of variables) can be found in =-=[Bax82]-=- or [BRSSZ01]. Recall that x ≺ z denotes that x is a child of z. A simple direct calculation gives that Rz,ℓ = � e−2βh x≺z F (Rx,ℓ−1), where F (a) ≡ Fβ(a) = a+e−2β e−2β . In particular, if τ is the al... |

270 |
Gibbs Measures and Phase Transitions, de Gruyter
- Georgii
- 1988
(Show Context)
Citation Context ...arge connected regions of equal spins tend to form. Example 2.2 Another famous example is the hard-core model (independent sets). This has been used in statistical physics as a model of lattice gases =-=[Geo88]-=-, and also in other areas such as the modeling of communication networks [Kel85]. Again there are just two spins S = {0, 1}, and we refer to a site as occupied if it has spin value 1 and unoccupied ot... |

261 | Approximate counting, uniform generation and rapidly mixing Markov chains
- Jerrum, Sinclair
- 1989
(Show Context)
Citation Context ...s that considers the progress made in one step of arbitrary instances of the chain. Other examples of this approach include bounds on the mixing time stemming from the conductance of the Markov chain =-=[SJ89]-=- or from the congestion of a flow along its paths [Sin92], as well as a series of papers [DF01, Mol02, Hay03, DFPV04] that analyze the Glauber dynamics for sampling proper colorings, where the argumen... |

234 |
Random walks on finite groups and rapidly mixing Markov chains
- Aldous
- 1983
(Show Context)
Citation Context ...η Ψ .sIn the literature, the dependency on ɛ is often omitted and the definition only considers a fixed ɛ, e.g., ɛ = 1 2e . The reason for this is the following well-known property of the mixing time =-=[Ald83]-=-: for any ɛ ≤ 1 1 e , τ(ɛ) ≤ τ( 2e 21 1 )⌈log( ɛ )⌉. Nevertheless, we include ɛ in the definition here because, as we will see next, we work with rates of decay in which the dependence on ɛ is sharper... |

193 | Beitrag zur Theorie des Ferromagnetismus - Ising - 1925 |

183 | Improved bounds for mixing rates of Markov chains and multicommodity flow
- Sinclair
- 1992
(Show Context)
Citation Context ...ary instances of the chain. Other examples of this approach include bounds on the mixing time stemming from the conductance of the Markov chain [SJ89] or from the congestion of a flow along its paths =-=[Sin92]-=-, as well as a series of papers [DF01, Mol02, Hay03, DFPV04] that analyze the Glauber dynamics for sampling proper colorings, where the argument relies on the fact that after a sufficient “burn-in” ti... |

181 | The statistical mechanics of lattice gases - Simon - 1993 |

175 | Sharp thresholds of graph properties and the k-SAT problem - Friedgut - 1998 |

149 | Path coupling: a technique for proving rapid mixing in Markov chains
- Bubley, Dyer
- 1997
(Show Context)
Citation Context ...ombinatorial in nature and involve a dynamical analysis similar to that carried out for analyzing mixing times of certain dynamics. We make heavy use of couplings, especially the path coupling method =-=[BD97]-=-. Our conditions for uniqueness essentially give the stronger property of weak spatial mixing (Definition 2.3). In addition, natural extensions of them imply strong spatial mixing (Definition 2.4) and... |

144 | Sur les inégalités de Sobolev logarithmiques, volume 10 of Panoramas et Synthèses. Société Mathématique de - Ané, Blachère, et al. - 2000 |

144 |
Some generalized orderdisorder transformations
- Potts, Domb
- 1952
(Show Context)
Citation Context ... all the sites in Ψ are unoccupied is assigned positive probability under all boundary conditions.sExample 2.3 A generalization of the Ising model to more than two spin values was introduced by Potts =-=[Pot]-=- and is known as the (ferromagnetic) Potts model. Here S = {1, 2, . . . , q} and the potentials are Ux,y(s1, s2) = −2βδs1,s2 , Ux(s) = 0. Thus the spin at each site can take one of q possible values, ... |

125 | The Potts model - Wu - 1982 |

112 | Lectures on Glauber dynamics for discrete spin models Lectures on probability theory and statistics
- Martinelli
- 1999
(Show Context)
Citation Context ...n Martinelli and Olivieri [MO94a, MO94b], who later obtained sharper results by working with a weaker spatial mixing assumption, and Cesi [Ces01], who recently simplified some of the proofs. See also =-=[Mar98]-=- for a review of related results. We shed new light on this equivalence by presenting new, simpler proofs of it. Furthermore, we explicitly point out the properties of the integer lattice used in the ... |

110 |
Stochastic models of computer communication systems
- Kelly
- 1985
(Show Context)
Citation Context ...xample is the hard-core model (independent sets). This has been used in statistical physics as a model of lattice gases [Geo88], and also in other areas such as the modeling of communication networks =-=[Kel85]-=-. Again there are just two spins S = {0, 1}, and we refer to a site as occupied if it has spin value 1 and unoccupied otherwise. A configuration therefore specifies a subset of occupied sites. The pot... |

93 | The random geometry of equilibrium phases. Phase transitions and critical phenomena (C
- Georgii, Häggström, et al.
- 2001
(Show Context)
Citation Context ...y elusive. We go on to describe some of the currently known facts regarding the Ising model with (+)-boundary at low temperatures. We start with the Gibbs distribution. It is known ([Dob96], and also =-=[GHM01]-=-) that the Gibbs distribution conditioned on the all-(+) boundary exhibits exponential decay of correlations as in Definition 2.5. Specifically, for all β > βc, there exist constants C and α > 0 depen... |

85 | Approach to equilibrium of Glauber dynamics in the one phase region. II. The general case
- Martinelli, Olivieri
- 1994
(Show Context)
Citation Context ...dynamics uniformly in the boundary condition (Theo77s78 rem 4.2). We state two theorems whose combination gives Theorem 4.2. The first theorem, whose proof uses ideas from the proof of Theorem 4.2 of =-=[MO94a]-=-, states that the strong spatial mixing assumption implies O(m log 2 n) coupling time of any monotone identity coupling, uniformly in the volume n and in the boundary configuration. The second theorem... |

73 |
Improved bounds for sampling colorings
- Vigoda
- 2000
(Show Context)
Citation Context ...reject”). There � exp(−HΘ (σ)) i exp(−HΘ (τ)) , 1 i are other examples of more sophisticated update rules which are specific to certain models (e.g., the update rule for proper colorings described in =-=[Vig00]-=-, or the one for independent sets [DG00] which we discuss in Section 3.4). Since (by property 1 of Definition 3.1) the spin of a given site may change only when updating a block that includes this sit... |

68 | On the purity of limiting Gibbs state for the Ising model on the Bethe lattice - Bleher, Ruiz, et al. - 1995 |

68 | On Markov chains for independent sets
- Dyer, Greenhill
- 2000
(Show Context)
Citation Context ...(τ)) , 1 i are other examples of more sophisticated update rules which are specific to certain models (e.g., the update rule for proper colorings described in [Vig00], or the one for independent sets =-=[DG00]-=- which we discuss in Section 3.4). Since (by property 1 of Definition 3.1) the spin of a given site may change only when updating a block that includes this site, we will often need to refer to the su... |

65 |
Lectures on finite Markov chains. Lectures on probability theory and statistics (Saint-Flour
- Saloff-Coste
- 1996
(Show Context)
Citation Context ...ate of decay of variance and entropy respectively, which implies that the mixing time is bounded in terms of these quantities. These classical facts are expressed in the following theorem (see, e.g., =-=[Sal97]-=-). Theorem 2.9 Let P be a Markov chain for µ τ Ψ , and let πmin stand for the minimum non-zero probability of a configuration under µ η Ψ . (Notice that πmin ≥ exp(−Cn) for some constant C that depend... |

59 | Absence of phase transition for antiferromagnetic Potts models via the Dobrushin uniqueness theorem
- Salas, Sokal
- 1997
(Show Context)
Citation Context ...e show uniqueness here is still larger than that given by the original Dobrushin condition, which is q > 2(max {b1, b2} + 1). (The calculation using the original Dobrushin condition was first done in =-=[SS97]-=-, and implies uniqueness for q > 2(b + 1) on any underlying graph of maximum degree b + 1. In order to use our theorems we need to specify a coupled update rule for the model. For this example, we use... |

56 |
Estimates of Semiinvariants for the Ising Model at Low Temperatures
- Dobrushin
- 1994
(Show Context)
Citation Context ...r proved to be very elusive. We go on to describe some of the currently known facts regarding the Ising model with (+)-boundary at low temperatures. We start with the Gibbs distribution. It is known (=-=[Dob96]-=-, and also [GHM01]) that the Gibbs distribution conditioned on the all-(+) boundary exhibits exponential decay of correlations as in Definition 2.5. Specifically, for all β > βc, there exist constants... |

51 | Phase Transitions on Non-amenable Graphs - Lyons - 2000 |

48 | The logarithmic Sobolev inequality for discrete spin systems on the lattice
- Stroock, Zegarlinski
- 1992
(Show Context)
Citation Context ...ondition is applicable only when the underlying graph of sites is the integer lattice Z d . Additional versions of the Dobrushin-Shlosman condition were given by others (e.g., Stroock and Zegarlinski =-=[SZ92]-=-), but still only in the context of Z d .sWe generalize the above conditions by considering both larger volumes and any underlying graph. Naturally, all such conditions require that the influence spin... |

47 |
Constructive criterion for the uniqueness of Gibbs field
- DOBRUSHIN, SHLOSMAN
- 1985
(Show Context)
Citation Context ...nt “burn-in” time, the chain is very unlikely to be in a “bad” configuration. Dobrushin type conditions The first results we present are generalizations of the Dobrushin [Dob70] and DobrushinShlosman =-=[DS85a]-=- conditions for uniqueness of the Gibbs measure. As their name suggests, conditions of this type were originally introduced as tools for establishing uniqueness of the infinite volume Gibbs measure (a... |

44 | Fast convergence of the Glauber dynamics for sampling independent sets, Random Structures and Algorithms 15 - Luby, Vigoda - 1999 |

43 |
Random growth in a tesselation
- Richardson
- 1973
(Show Context)
Citation Context ...f propagation of information originally appeared in [SZ92], and similar bounds can also be found in [Mar98, KMP01]. The quantitative analysis in the argument in fact goes back to the Richardson model =-=[Ric73]-=-. Our version below applies to the Glauber dynamics on general graphs (as in [KMP01]), rather than just Z d . Lemma 4.4 Consider an arbitrary spin system on an arbitrary graph of maximum degree b + 1.... |

40 | The Ising model and percolation on trees and tree-like graphs - Lyons - 1989 |

39 | F.: The spectral gap for a Glauber-type dynamics in a continuous gas - Bertini, Cancrini, et al. - 2002 |

39 | Information flow on trees
- Mossel, Peres
- 2004
(Show Context)
Citation Context ...e uniqueness region (but not for arbitrary β) for all combinations of b and q. Second, because of an intimate connection between the free boundary case and socalled “reconstruction problems” on trees =-=[Mos02]-=- (in which the edges are noisy channels and the goal is to reconstruct a value transmitted from the root), we obtain an alternative proof of the best known value of the noise parameter under which rec... |

38 |
Glauber dynamics on trees and hyperbolic graphs, Probab. Theory Related Fields 131
- Berger, Kenyon, et al.
- 2005
(Show Context)
Citation Context ...zation of the mixing time for the Ising model on the square lattice described above.) An appropriate notion of temporal mixing is also known to imply a certain decay of correlations in general graphs =-=[KMP01]-=-. The motivation for further studying temporal-spatial correspondences in this dissertation stems from a need for progress on three levels. First, we seek a better understanding of the mathematical pr... |

38 | Fast mixing for independent sets, colorings, and other models on trees. Random Structures Algorithms 31 - MARTINELLI, SINCLAIR, et al. - 2007 |

38 | private communication
- Fischer
- 2005
(Show Context)
Citation Context ...e conclude this section with the detailed results proven in this chapter for the 2 Actually [KMP01] proves this only for sufficiently high temperatures, but the argument can be extended to all β < β1 =-=[PW02]-=-.s−(b−1) T = 1/β Figure 5.1: Curve of critical field hc(β). The Gibbs measure is unique above the curve. Ising model on T b . Theorem 5.2 For any fixed b, the Glauber dynamics on the n-vertex b-ary tr... |

37 | A note on the extremality of the disordered state for the Ising model on the Bethe lattice - Ioffe - 1996 |

35 |
Rapid convergence to equilibrium of stochastic Ising models in the Dobrushin Shlosman regime, in: Percolation theory and ergodic theory of in particle systems
- Aizenman, Holley
- 1987
(Show Context)
Citation Context ... exponential decay (with distance) of the influence of the boundary condition on the equilibrium distribution. The first results in this direction were given by Holley [Hol85] and Aizenman and Holley =-=[AH87]-=-, followed by Zegarlinski [Zeg90] and culminating in the work of Stroock and Zegarlinski [SZ92], who were the first to establish the above equivalence in full. We further mention Martinelli and Olivie... |

35 |
A Non-Markovian Coupling for Randomly Sampling Colorings
- Hayes, Vigoda
- 2003
(Show Context)
Citation Context ...e required number of colors further for general graphs, under the assumption that the maximum degree is Ω(log n). The current state of the art requires q ≥ (1 + ɛ)(b + 1), for arbitrarily small ɛ > 0 =-=[HV03]-=-, but these results do not apply in our setting where the degree b + 1 is fixed.snotice that the dynamics is connected for q ≥ b + 3 (on any graph of maximum degree b + 1, as was mentioned in Section ... |

34 | Survey propagation: an algorithm for satisfiability - Braunstein, Mézard, et al. |

34 | Randomly Colouring Graphs with Lower Bounds on - Dyer, Frieze |

33 |
A uniqueness condition for gibbs measures, with application to the 2-dimensional ising antiferromagnet
- Berg
- 1993
(Show Context)
Citation Context ...namical process. In this section we give a lemma of this sort following an argument explained to us by van den Berg, based on the idea of paths of disagreement (also known as disagreement percolation =-=[Ber93]-=-). The idea of bounding the speed of propagation of information originally appeared in [SZ92], and similar bounds can also be found in [Mar98, KMP01]. The quantitative analysis in the argument in fact... |

33 | Schonmann For 2-D lattice spin systems weak mixing implies strong mixing
- Martinelli, Olivieri, et al.
- 1994
(Show Context)
Citation Context ...ong spatial mixing hold, while for β > ˆ βc both fail. It is worth mentioning here that, in the special case of the Ising model on the square lattice Z 2 , the critical ˆ βc mentioned above coincides =-=[MOS94]-=- with the critical inverse temperature βc at which a phase transition occurs in the infinite volume limit, namely, for β < βc there exists a unique infinite volume Gibbs measure while for β > βc there... |

31 | The Glauber Dynamics on Colorings of a Graph with High Girth and - Molloy - 2002 |

30 | Log-Sobolev inequalities and sampling from log-concave distributions - Frieze, Kannan |

26 | A note on the Glauber dynamics for sampling independent sets - Vigoda |

23 |
The random-cluster model on a homogeneous tree. Probab. Theory and Relat
- Häggström
- 1996
(Show Context)
Citation Context ... at a critical β = β0, which depends on b and q, so that for β > β0 (and indeed for β ≥ β0 when q > 2) there are multiple phases. This value β0 does not in general have a closed form, but it is known =-=[Hag96]-=- that β0 < 1 2 ln( b+q−1 b−1 ) for all q > 2. (For q = 2, this value is exactly β0 for the Ising model as quoted earlier.)s132 Little is known about the Glauber dynamics for the Potts model on trees, ... |

23 | Amenability and phase transition in the Ising model - Jonasson, Steif - 1999 |

22 | Mixing in time and space for lattice spin systems: a combinatorial view, Rand Structs Algos 24
- Dyer, Sinclair, et al.
- 2004
(Show Context)
Citation Context ...d for publication as [Wei03]. The combinatorial proofs for the known equivalence between temporal and spatial mixing on the integer lattice are given in Chapter 4; this chapter is based on joint work =-=[DSVW02]-=- with Martin Dyer, Alistair Sinclair and Eric Vigoda. In Chapter 5 we present our new boundary-specific equivalence for systems on trees, as well as its applications; this chapter is based on joint wo... |

22 | Uniqueness of uniform random colorings of regular trees
- Jonasson
- 2006
(Show Context)
Citation Context ...eness for colorings on a tree is known to hold for a wider range of parameters. For example, when b1 = b2 (the tree is regular of degree b1 + 1), the Gibbs measure is unique if and only if q > b1 + 1 =-=[Jon02]-=-. However, the range of parameters for which we show uniqueness here is still larger than that given by the original Dobrushin condition, which is q > 2(max {b1, b2} + 1). (The calculation using the o... |

22 | Glauber dynamics on trees: boundary conditions and mixing time
- Martinelli, Sinclair, et al.
(Show Context)
Citation Context ... lattice is given in Chapter 6 and appears only in this dissertation. The theory of decomposing variance and entropy is presented in Appendix A; many of the new proofs in this appendix are taken from =-=[MSW03]-=-. Appendix B contains a proof deferred from Chapter 5 and is also taken from [MSW03]. 9s10 Chapter 2 Preliminaries In this chapter we set the basis for the discussion in the rest of this thesis by def... |

21 | Quasi-factorization of the entropy and logarithmic Sobolev inequalities for Gibbs random fields, Probab. Theory Related Fields
- Cesi
- 2001
(Show Context)
Citation Context ... to establish the above equivalence in full. We further mention Martinelli and Olivieri [MO94a, MO94b], who later obtained sharper results by working with a weaker spatial mixing assumption, and Cesi =-=[Ces01]-=-, who recently simplified some of the proofs. See also [Mar98] for a review of related results. We shed new light on this equivalence by presenting new, simpler proofs of it. Furthermore, we explicitl... |

21 |
Completely analytical Gibbs fields
- DOBRUSHIN, SHLOSMAN
- 1985
(Show Context)
Citation Context ... the proofs of Theorems 3.3 and 3.5.) • The fact that stronger conditions such as these indicated here imply optimal temporal mixing and strong spatial mixing was already proved for systems on Z d in =-=[DS85b]-=- and [SZ92]. • The converse for systems on Z d is also known [DS85b, SZ92]. In particular, if strong spatial mixing holds then the conditions in Corollaries 3.12 and 3.14 hold w.r.t. the heat-bath upd... |