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22
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 23 (7 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).
Quasifactorization of the entropy and logarithmic Sobolev inequalities for Gibbs random fields
, 2001
"... . We show that the entropy functional exhibits a quasi{factorization property with respect to a pair of weakly dependent {algebras. As an application we give a simple proof that the Dobrushin and Shlosman's complete analyticity condition, for a Gibbs specication with nite range summable interaction, ..."
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Cited by 20 (0 self)
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. We show that the entropy functional exhibits a quasi{factorization property with respect to a pair of weakly dependent {algebras. As an application we give a simple proof that the Dobrushin and Shlosman's complete analyticity condition, for a Gibbs specication with nite range summable interaction, implies uniform logarithmic Sobolev inequalities. This result has been previously proven using several dierent techniques. The advantage of our approach is that it relies almost entirely on a general property of the entropy, while very little is assumed on the Dirichlet form. No topology is introduced on the single spin space, thus discrete and continuous spins can be treated in the same way. Key Words: Entropy, Logarithmic Sobolev inequalities, Gibbs measures Mathematics Subject Classication 2000: 82B20, 82C20, 39B62 v1.0 1. Introduction Logarithmic Sobolev inequalities have been introduced in [Gr1] where it has been shown that Z R d f 2 (x) log jf(x)j d (dx) Z R d jr...
The logarithmic Sobolev constant of Kawasaki dynamics under a mixing condition revisited
, 2000
"... . We consider a conservative stochastic spin exchange dynamics reversible with respect to the canonical Gibbs measure of a lattice gas model. We assume that the corresponding grand canonical measure satises a suitable strong mixing condition. Following previous work by two of us for the spectral gap ..."
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Cited by 9 (1 self)
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. We consider a conservative stochastic spin exchange dynamics reversible with respect to the canonical Gibbs measure of a lattice gas model. We assume that the corresponding grand canonical measure satises a suitable strong mixing condition. Following previous work by two of us for the spectral gap, we provide an alternative and quite natural, from the physical point of view, proof of the well known result of Yau stating that the logarithmic Sobolev constant in a box of side L grows like L 2 . Key Words: Kawasaki dynamics, logarithmic Sobolev constant, equivalence of ensembles, concentration inequalities. Mathematics Subject Classication: 82B44, 82C22, 82C44, 60K35 v1.02 1. Introduction In this paper we provide a new, independent proof of the famous result of Yau [Y] which states that the logarithmic Sobolev constant of a spin exchange dynamics in a box of side L of Z d , reversible w.r.t. the canonical Gibbs measure of a nite range lattice gas, grows like L 2 , provided t...
ISOPERIMETRIC INEQUALITIES AND MIXING TIME FOR A RANDOM WALK ON A RANDOM POINT PROCESS
, 2006
"... Abstract. We consider the random walk on a simple point process on R d, d � 2, whose jump rates decay exponentially in the α–power of jump length. The case α = 1 corresponds to the phonon–induced variable–range hopping in disordered solids in the regime of strong Anderson localization. Under mild as ..."
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Cited by 6 (5 self)
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Abstract. We consider the random walk on a simple point process on R d, d � 2, whose jump rates decay exponentially in the α–power of jump length. The case α = 1 corresponds to the phonon–induced variable–range hopping in disordered solids in the regime of strong Anderson localization. Under mild assumptions on the point process, we show for α ∈ (0, d) that the random walk confined to a cubic box of side L has a.s. Cheeger constant of order at least L −1 and mixing time of order L 2. For the Poisson point process we prove that at α = d there is a transition from diffusive to subdiffusive behavior of the random walk. Key words: Random walk in random environment, point process, isoperimetric inequality, mixing time, isoperimetric profile, percolation.
Kinetically constrained spin models
"... ABSTRACT. We analyze the density and size dependence of the relaxation time for kinetically constrained spin models (KCSM) intensively studied in the physical literature as simple models sharing some of the features of a glass transition. KCSM are interacting particle systems on Z d with Glauberlik ..."
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Cited by 5 (0 self)
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ABSTRACT. We analyze the density and size dependence of the relaxation time for kinetically constrained spin models (KCSM) intensively studied in the physical literature as simple models sharing some of the features of a glass transition. KCSM are interacting particle systems on Z d with Glauberlike dynamics, reversible w.r.t. a simple product i.i.d Bernoulli(p) measure. The essential feature of a KCSM is that the creation/destruction of a particle at a given site can occur only if the current configuration of empty sites around it satisfies certain constraints which completely define each specific model. No other interaction is present in the model. From the mathematical point of view, the basic issues concerning positivity of the spectral gap inside the ergodicity region and its scaling with the particle density p remained open for most KCSM (with the notably exception of the East model in d = 1 [3]). Here for the first time we: i) identify the ergodicity region by establishing a connection with an associated bootstrap percolation model; ii) develop a novel multiscale approach which proves positivity of the spectral gap in the whole ergodic region; iii) establish, sometimes optimal, bounds on the behavior of the spectral gap near the boundary of the ergodicity region and iv) establish pure exponential decay for the persistence function (see below). Our techniques are flexible enough to allow a variety of constraints and our findings disprove certain conjectures which appeared in the physical literature on the basis of numerical simulations.
Spectral gap estimates for interacting particle systems via a Bochner type inequality
 J. Funct. Anal
, 2006
"... We develop a general technique, based on the Bakry–Emery approach, to estimate spectral gaps of a class of Markov operator. We apply this technique to various interacting particle systems. In particular, we give a simple and short proof of the diffusive scaling of the spectral gap of the Kawasaki mo ..."
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Cited by 4 (0 self)
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We develop a general technique, based on the Bakry–Emery approach, to estimate spectral gaps of a class of Markov operator. We apply this technique to various interacting particle systems. In particular, we give a simple and short proof of the diffusive scaling of the spectral gap of the Kawasaki model at high temperature. Similar results are derived for Kawasakitype dynamics in the lattice without exclusion, and in the continuum. New estimates for Glaubertype dynamics are also obtained. 1 1
Equilibrium Kawasaki dynamics of continuous particle systems
 INFIN. DIMENS. ANAL. QUANTUM PROBAB. RELAT. TOP
, 2007
"... We construct a new equilibrium dynamics of infinite particle systems in a Riemannian manifold X. This dynamics is an analog of the Kawasaki dynamics of lattice spin systems. The Kawasaki dynamics now is a process where interacting particles randomly hop over X. We establish conditions on the a prior ..."
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Cited by 3 (2 self)
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We construct a new equilibrium dynamics of infinite particle systems in a Riemannian manifold X. This dynamics is an analog of the Kawasaki dynamics of lattice spin systems. The Kawasaki dynamics now is a process where interacting particles randomly hop over X. We establish conditions on the a priori explicitly given symmetrizing measure and the generator of this dynamics, under which a corresponding conservative Markov processes exists. We also outline two types of scaling limit of the equilibrium Kawasaki dynamics: one leading to an equilibrium Glauber dynamics in continuum (a birthanddeath process), and the other leading to a diffusion dynamics of interacting particles (in particular, the gradient stochastic dynamics).
A Combinatorial Proof of Tree Decay of Semiinvariants
"... We consider nite range Gibbs elds and provide a purely combinatorial proof of the exponential tree decay of semi{invariants, supposing that the logarithm of the partition function can be expressed as a sum of suitable local functions of the boundary conditions. This hypothesis holds for completely ..."
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Cited by 1 (1 self)
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We consider nite range Gibbs elds and provide a purely combinatorial proof of the exponential tree decay of semi{invariants, supposing that the logarithm of the partition function can be expressed as a sum of suitable local functions of the boundary conditions. This hypothesis holds for completely analytical Gibbs elds; in this context the tree decay of semi{invariants has been proven via analyticity arguments. However the combinatorial proof given here can be applied also to the more complicated case of disordered systems in the so called Griths' phase when analyticity arguments fail.
A note on equilibrium Glauber and Kawasaki dynamics for fermion point processes
, 2007
"... We construct two types of equilibrium dynamics of infinite particle systems in a locally compact Polish space X, for which certain fermion point processes are invariant. The Glauber dynamics is a birthanddeath process in X, while in the case of the Kawasaki dynamics interacting particles randomly ..."
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Cited by 1 (1 self)
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We construct two types of equilibrium dynamics of infinite particle systems in a locally compact Polish space X, for which certain fermion point processes are invariant. The Glauber dynamics is a birthanddeath process in X, while in the case of the Kawasaki dynamics interacting particles randomly hop over X. We establish conditions on generators of both dynamics under which corresponding conservative Markov processes exist.