In this work we show that for every @math , such that for all @math where the parameters of the model do not depend on @math . They also provide a rare example where one can prove a polynomial time mixing of Gibbs sampler in a situation where the actual mixing time is slower than @math . Our proof exploits in novel ways the local treelike structure of Erd o s-R 'enyi random graphs, comparison and block dynamics arguments and a recent result of Weitz. Our results extend to much more general families of graphs which are sparse in some average sense and to much more general interactions. In particular, they apply to any graph for which every vertex @math of the graph has a neighborhood @math of radius @math in which the induced sub-graph is a tree union at most @math edges and where for each simple path in @math the sum of the vertex degrees along the path is @math . Moreover, our result apply also in the case of arbitrary external fields and provide the first FPRAS for sampling the Ising distribution in this case. We finally present a non Markov Chain algorithm for sampling the distribution which is effective for a wider range of parameters. In particular, for @math it applies for all external fields and @math , where @math is the critical point for decay of correlation for the Ising model on @math .
It is natural to conjecture that properties of the Ising model on the branching process with @math offspring distribution determines the mixing time of the dynamics on @math . In particular, it is natural to conjecture that the critical point for uniqueness of Gibbs measures plays a fundamental role @cite_19 @cite_12 as results of similar flavor were recently obtained for the hard-core model on random bi-partite @math regular graphs @cite_14 .
@cite_19 @cite_14 @cite_12
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In this work we show that for every @math , such that for all @math where the parameters of the model do not depend on @math . They also provide a rare example where one can prove a polynomial time mixing of Gibbs sampler in a situation where the actual mixing time is slower than @math . Our proof exploits in novel ways the local treelike structure of Erd o s-R 'enyi random graphs, comparison and block dynamics arguments and a recent result of Weitz. Our results extend to much more general families of graphs which are sparse in some average sense and to much more general interactions. In particular, they apply to any graph for which every vertex @math of the graph has a neighborhood @math of radius @math in which the induced sub-graph is a tree union at most @math edges and where for each simple path in @math the sum of the vertex degrees along the path is @math . Moreover, our result apply also in the case of arbitrary external fields and provide the first FPRAS for sampling the Ising distribution in this case. We finally present a non Markov Chain algorithm for sampling the distribution which is effective for a wider range of parameters. In particular, for @math it applies for all external fields and @math , where @math is the critical point for decay of correlation for the Ising model on @math . 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 β < 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 β, and arbitrary external fields are bounded by Cnlogn. 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 dtanh β < 1, with high probability over the Erdős–Renyi random graph G(n,d n), it holds that the mixing time of Gibbs samplers is n1+Θ(1 loglogn). Both results are tight, as it is known that the mixing time for random regular and Erdős–Renyi random graphs is, with high probability, exponential in n when (d−1) tanh β> 1, and dtanh β>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. We prove that for finite range discrete spin systems on the two dimensional latticeZ2, the (weak) mixing condition which follows, for instance, from the Dobrushin-Shlosman uniqueness condition for the Gibbs state implies a stronger mixing property of the Gibbs state, similar to the Dobrushin-Shlosman complete analyticity condition, but restricted to all squares in the lattice, or, more generally, to all sets multiple of a large enough square. The key observation leading to the proof is that a change in the boundary conditions cannot propagate either in the bulk, because of the weak mixing condition, or along the boundary because it is one dimensional. As a consequence we obtain for ferromagnetic Ising-type systems proofs that several nice properties hold arbitrarily close to the critical temperature; these properties include the existence of a convergent cluster expansion and uniform boundedness of the logarithmic Sobolev constant and rapid convergence to equilibrium of the associated Glauber dynamics on nice subsets ofZ2, including the full lattice. We study supercritical branching Brownian motion on the real line starting at the origin and with constant drift @math . At the point @math , we add an absorbing barrier, i.e. individuals touching the barrier are instantly killed without producing offspring. It is known that there is a critical drift @math , such that this process becomes extinct almost surely if and only if @math . In this case, if @math denotes the number of individuals absorbed at the barrier, we give an asymptotic for @math as @math goes to infinity. If @math and the reproduction is deterministic, this improves upon results of [L. Addario-Berry and N. Broutin (2009), http: arxiv.org abs 0908.1083v1 ] and [E. A " dekon (2009), http: arxiv.org abs 0911.0877v1 ] on a conjecture by David Aldous about the total progeny of a branching random walk. The main technique used in the proofs is analysis of the generating function of @math near its singular point @math , based on classical results on some complex differential equations.