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Logarithmic Sobolev inequality and finite markov chains
, 1996
"... This is an expository paper on the use of logarithmic Sobolev inequalities for bounding rates of convergence of Markov chains on finite state spaces to their stationary distributions. Logarithmic Sobolev inequalities complement eigenvalue techniques and work for nonreversible chains in continuous ti ..."
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Cited by 180 (14 self)
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This is an expository paper on the use of logarithmic Sobolev inequalities for bounding rates of convergence of Markov chains on finite state spaces to their stationary distributions. Logarithmic Sobolev inequalities complement eigenvalue techniques and work for nonreversible chains in continuous time. Some aspects of the theory simplify considerably with finite state spaces and we are able to give a selfcontained development. Examples of applications include the study of a Metropolis chain for the binomial distribution, sharp results for natural chains on the box of side n in d dimensions and improved rates for exclusion processes. We also show that for most rregular graphs the logSobolev constant is of smaller order than the spectral gap. The logSobolev constant of the asymmetric twopoint space is computed exactly as well as the logSobolev constant of the complete graph on n points.
Approach to Equilibrium of Glauber Dynamics In the One Phase Region. II: The General Case
, 1994
"... . We develop a new method, based on renormalization group ideas (block decimation procedure), to prove, under an assumption of strong mixing in a finite cube o , a Logarithmic Sobolev Inequality for the Gibbs state of a discrete spin system. As a consequence we derive the hypercontractivity of the ..."
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Cited by 120 (16 self)
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. We develop a new method, based on renormalization group ideas (block decimation procedure), to prove, under an assumption of strong mixing in a finite cube o , a Logarithmic Sobolev Inequality for the Gibbs state of a discrete spin system. As a consequence we derive the hypercontractivity of the Markov semigroup of the associated Glauber dynamics and the exponential convergence to equilibrium in the uniform norm in all volumes "multiples" of the cube o . Work partially supported by grant SC1CT910695 of the Commission of European Communities 25=aprile=1997 [1] 0:1 Section 1. Preliminaries, Definitions and Results In this paper we analyze the problem of the approach to equilibrium for a general, not necessarily ferromagnetic, Glauber dynamics, i.e. a single spin flip stochastic dynamics reversible with respect to the Gibbs measure of a classical discrete spin system with finite range, translation invariant interaction. We prove that, if the Gibbs measure satisfies a Strong Mix...
Mixing times of lozenge tiling and card shuffling Markov chains
, 1997
"... Abstract. We show how to combine Fourier analysis with coupling arguments to bound the mixing times of a variety of Markov chains. The mixing time is the number of steps a Markov chain takes to approach its equilibrium distribution. One application is to a class of Markov chains introduced by Luby, ..."
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Cited by 102 (1 self)
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Abstract. We show how to combine Fourier analysis with coupling arguments to bound the mixing times of a variety of Markov chains. The mixing time is the number of steps a Markov chain takes to approach its equilibrium distribution. One application is to a class of Markov chains introduced by Luby, Randall, and Sinclair to generate random tilings of regions by lozenges. For an ℓ×ℓ region we bound the mixing time by O(ℓ 4 log ℓ), which improves on the previous bound of O(ℓ 7), and we show the new bound to be essentially tight. In another application we resolve a few questions raised by Diaconis and SaloffCoste by lower bounding the mixing time of various cardshuffling Markov chains. Our lower bounds are within a constant factor of their upper bounds. When we use our methods to modify a pathcoupling analysis of Bubley and Dyer, we obtain an O(n 3 log n) upper bound on the mixing time of the KarzanovKhachiyan Markov chain for linear extensions. 1.
The spectral gap for a Glaubertype dynamics in a continuous gas
, 2000
"... . We consider a continuous gas in a d dimensional rectangular box with a nite range, positive pair potential, and we construct a Markov process in which particles appear and disappear with appropriate rates so that the process is reversible w.r.t. the Gibbs measure. If the thermodynamical paramenter ..."
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Cited by 54 (8 self)
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. We consider a continuous gas in a d dimensional rectangular box with a nite range, positive pair potential, and we construct a Markov process in which particles appear and disappear with appropriate rates so that the process is reversible w.r.t. the Gibbs measure. If the thermodynamical paramenters are such that the Gibbs specication satises a certain mixing condition, then the spectral gap of the generator is strictly positive uniformly in the volume and boundary condition. The required mixing condition holds if, for instance, there is a convergent cluster expansion. Key Words: Spectral gap, Gibbs measures, continuous systems, birth and death processes Mathematics Subject Classication: 82C21, 60K35, 82C22, 60J75 This work was partially supported by GNAFA and by \Conanziamento Murst" v1.4 1. Introduction We consider a continuous gas in a bounded volume R d , distributed according the Gibbs probability measure associated to a nite range pair potential '. The Gibbs measu...
Universality of random matrices and local relaxation flow
"... We consider N × N symmetric random matrices where the probability distribution for each matrix element is given by a measure ν with a subexponential decay. We prove that the eigenvalue spacing statistics in the bulk of the spectrum for these matrices and for GOE are the same in the limit N → ∞. Our ..."
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Cited by 54 (12 self)
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We consider N × N symmetric random matrices where the probability distribution for each matrix element is given by a measure ν with a subexponential decay. We prove that the eigenvalue spacing statistics in the bulk of the spectrum for these matrices and for GOE are the same in the limit N → ∞. Our approach is based on the study of the Dyson Brownian motion via a related new dynamics, the local relaxation flow.
For 2D lattice spin systems Weak Mixing Implies Strong Mixing
"... . We prove that for finite range discrete spin systems on the two dimensional lattice Z 2 , the (weak) mixing condition which follows, for instance, from the DobrushinShlosman uniqueness condition for the Gibbs state implies a stronger mixing property of the Gibbs state, similar to the DobrushinS ..."
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Cited by 45 (7 self)
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. We prove that for finite range discrete spin systems on the two dimensional lattice Z 2 , the (weak) mixing condition which follows, for instance, from the DobrushinShlosman uniqueness condition for the Gibbs state implies a stronger mixing property of the Gibbs state, similar to the DobrushinShlosman complete analiticity 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 Isingtype 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 assoc...
Glauber dynamics on trees: boundary conditions and mixing time
 Comm. Math. Phys
"... We give the first comprehensive analysis of the effect of boundary conditions on the mixing time of the Glauber dynamics in the socalled Bethe approximation. Specifically, we show that spectral gap and the logSobolev constant of the Glauber dynamics for the Ising model on an nvertex regular tree ..."
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Cited by 32 (10 self)
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We give the first comprehensive analysis of the effect of boundary conditions on the mixing time of the Glauber dynamics in the socalled Bethe approximation. Specifically, we show that spectral gap and the logSobolev constant of the Glauber dynamics for the Ising model on an nvertex regular tree with (+)boundary are bounded below by a constant independent of n at all temperatures and all external fields. This implies that the mixing time is O(log n) (in contrast to the free boundary case, where it is not bounded by any fixed polynomial at low temperatures). In addition, our methods yield simpler proofs and stronger results for the spectral gap and logSobolev constant in the regime where there are multiple phases but the mixing time is insensitive to the boundary condition. Our techniques also apply to a much wider class of models, including those with hardcore constraints like the antiferromagnetic Potts model at zero temperature (proper colorings) and the hard–core lattice gas (independent sets).
Logarithmic Sobolev Inequality for Lattice Gases with Mixing Conditions
, 1995
"... Let ¯ gc L; denote the grand canonical Gibbs measure of a lattice gas in a cube of size L with the chemical potential and a fixed boundary condition. Let ¯ c L;n be the corresponding canonical measure defined by conditioning ¯ gc L; on P x2 j x = n. Consider the lattice gas dynamics ..."
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Cited by 29 (0 self)
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Let ¯ gc L; denote the grand canonical Gibbs measure of a lattice gas in a cube of size L with the chemical potential and a fixed boundary condition. Let ¯ c L;n be the corresponding canonical measure defined by conditioning ¯ gc L; on P x2 j x = n. Consider the lattice gas dynamics for which each particle performs random walk with rates depending on nearby particles. The rates are chosen such that, for every n and L fixed, ¯ c L ;n is a reversible measure. Suppose that the DobrushinShlosman mixing conditions holds for ¯ L; for all chemical potentials 2 R I . We prove that R f log fd¯ c L ;n const.L 2 D( p f ) for any probability density f with respect to ¯ c L;n ; here the constant is independent of n or L and D denotes the Dirichlet form of the dynamics. The dependence on L is optimal. Keywords: DobrushinShlosman mixing conditions, Interacting random walks, Lattice gas dynamics, Logarithmic Sobolev inequality Research partially supported by U...
On the Two Dimensional Dynamical Ising Model In the Phase Coexistence Region
 J. Stat. Phys
, 1994
"... . We consider a Glauber dynamics reversible with respect to the two dimensional Ising model in a finite square of side L, in the absence of an external field and at large inverse temperature fi. We first consider the gap in the spectrum of the generator of the dynamics in two different cases: with p ..."
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Cited by 28 (10 self)
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. We consider a Glauber dynamics reversible with respect to the two dimensional Ising model in a finite square of side L, in the absence of an external field and at large inverse temperature fi. We first consider the gap in the spectrum of the generator of the dynamics in two different cases: with plus and open boundary condition. We prove that, when the symmetry under global spin flip is broken by the boundary conditions, the gap is much larger than the case in which the symmetry is present. For this latter we compute exactly the asymptotics of \Gamma 1 fiL log(gap) as L ! 1 and show that it coincides with the surface tension along one of the coordinat axes. As a consequence we are able to study quite precisely the large deviations in time of the magnetization and to obtain an upper bound on the spinspin time correlation in the infinite volume plus phase. Our results establish a connection between the dynamical large deviations and those of the equilibrium Gibbs measure studied by...