Combinatorial criteria for uniqueness of Gibbs measures

Weitz, Dror

doi:10.1002/rsa.20073

Cited by 60 publications

(80 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We now show that spatial mixing in (12) and stabilizing graph in (13) imply weak stabilization in (9). …”

Section: Sufficient Conditions For Weak Stabilizationmentioning

confidence: 63%

“…Lemma 2 (Stabilization Regime Through Percolation): If G(P λfmax ; 1, ρ(·)) does not percolate, i.e., does not contain a giant component a.s. then the functional ξ of the form in (7) is weakly stabilizing according to (9) and hence, we have existence of limits in Theorem 1.…”

Section: B Stabilization Under Growing Maximum Degreementioning

confidence: 75%

“…Definition 3 (Almost-Sure (Weak) Spatial Mixing [9]): A Gibbs process P[Y V ] on a locally finite node set V exhibits weak spatial mixing a.s., when for measurement Y v at every point v ∈ V and any node subset V ⊂ V, we have a.s., (12) for any two feasible configurations y V , z V ∈ Y |V| , such that lim s→∞ δ(s) = 0. Hence, spatial mixing implies correlation decay over long distances since the influence of a node on another node decreases with their graph distance.…”

Section: A Spatial Mixingmentioning

confidence: 99%

“…Formally, the (Dobrushin) influence coefficient of t a node j on node i in a node set V is [9], [12],…”

Section: A Stabilization Under Finite Maximum Degreementioning

confidence: 99%

“…First, we establish conditions for existence of limits and derive the limits for a general class of functions of spatial graphical models on random Euclidean graphs. Second, we relate these conditions to notions of correlation decay of the Gibbs measure 1 (spatial mixing) [9] and stabilization property of the graph of the Gibbs measure [7], [8]. Third, we study specific graphs and models where these conditions hold.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Limit laws for random spatial graphical models

Anandkumar¹,

Yukich

Willsky³

2010

2010 IEEE International Symposium on Information Theory

View full text Add to dashboard Cite

Abstract-We consider spatial graphical models on random Euclidean points, applicable for data in sensor and social networks. We establish limit laws for general functions of the graphical model such as the mean value, the entropy rate etc. as the number of nodes goes to infinity under certain conditions. These conditions require the corresponding Gibbs measure to be spatially mixing and for the random graph of the model to satisfy a certain localization property known as stabilization. Graphs such the k nearest neighbor graph and the geometric disc graph belong to the class of stabilizing graphs. Intuitively, these conditions require the data at each node not to have strong dependence on data and positions of nodes far away. Finally, it is shown that spatial mixing of the Gibbs measure on a random graph holds when a suitably defined degree-dependent (but otherwise independent) node percolation does not have a giant component.

show abstract

“…We now show that spatial mixing in (12) and stabilizing graph in (13) imply weak stabilization in (9). …”

Section: Sufficient Conditions For Weak Stabilizationmentioning

confidence: 63%

Section: B Stabilization Under Growing Maximum Degreementioning

confidence: 75%

Section: A Spatial Mixingmentioning

confidence: 99%

“…Formally, the (Dobrushin) influence coefficient of t a node j on node i in a node set V is [9], [12],…”

Section: A Stabilization Under Finite Maximum Degreementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Limit laws for random spatial graphical models

Anandkumar¹,

Yukich

Willsky³

2010

2010 IEEE International Symposium on Information Theory

View full text Add to dashboard Cite

show abstract

Fast mixing for independent sets, colorings, and other models on trees

Martinelli

Sinclair

Weitz

2006

Random Struct Algorithms

Self Cite

108

165

View full text Add to dashboard Cite

We study the mixing time of the Glauber dynamics for general spin systems on bounded-degree trees, including the Ising model, the hard-core model (independent sets) and the antiferromagnetic Potts model at zero temperature (colorings). We generalize a framework, developed in our recent paper [18] in the context of the Ising model, for establishing mixing time O(n log n), which ties this property closely to phase transitions in the underlying model. We use this framework to obtain rapid mixing results for several models over a significantly wider range of parameter values than previously known, including situations in which the mixing time is strongly dependent on the boundary condition.

show abstract

Counting without sampling: Asymptotics of the log‐partition function for certain statistical physics models

Bandyopadhyay¹,

Gamarnik²

2008

Random Struct Algorithms

123

View full text Add to dashboard Cite

ABSTRACT:In this article we propose new methods for computing the asymptotic value for the logarithm of the partition function (free energy) for certain statistical physics models on certain type of finite graphs, as the size of the underlying graph goes to infinity. The two models considered are the hard-core (independent set) model when the activity parameter λ is small, and also the Potts (q-coloring) model. We only consider the graphs with large girth. In particular, we prove that asymptotically the logarithm of the number of independent sets of any r-regular graph with large girth when rescaled is approximately constant if r ≤ 5. For example, we show that every 4-regular n-node graph with large girth has approximately (1.494 · · · ) n -many independent sets, for large n. Further, we prove that for every r-regular graph with r ≥ 2, with n nodes and large girth, the number of proper q ≥ r + 1 colorings is approximatelyn , for large n. We also show that these results hold for random regular graphs with high probability (w.h.p.) as well.As a byproduct of our method we obtain simple algorithms for the problem of computing approximately the logarithm of the number of independent sets and proper colorings, in low degree COUNTING WITHOUT SAMPLING 453graphs with large girth. These algorithms are deterministic and use certain correlation decay properties for the corresponding Gibbs measures, and its implications to uniqueness of the Gibbs measures on the infinite trees, as well as some simple cavity trick which is well known in the physics and the Markov chain sampling literature.

show abstract

Combinatorial criteria for uniqueness of Gibbs measures

Cited by 60 publications

References 25 publications

Limit laws for random spatial graphical models

Limit laws for random spatial graphical models

Fast mixing for independent sets, colorings, and other models on trees

Counting without sampling: Asymptotics of the log‐partition function for certain statistical physics models

Contact Info

Product

Resources

About