Abstract. We prove new inequalities implying exponential decay of relative entropy functionals for a class of Zero–Range processes on the complete graph. We first consider the case of uniformly increasing rates, where we use a discrete version of the Bakry– Emery criterium to prove spectral gap and entropy dissipation estimates, uniformly over the number of particles and the number of vertices. We then study the standard case of possibly oscillating but roughly linearly increasing rates. Here the uniform entropy dissipation estimate is obtained by an adaptation of the martingale approach.

We prove that the logarithmic Sobolev constant for zero-range processes in a box of diameter L grows as L 2. 1. Introduction. Let Λ be a cube in Zd, and c:N → [0,+∞) be a function such that c(0) = 0 and c(n)> 0 for every n> 0. The zero-range process associated to c(·) is a stochastic system of moving particles in Λ, which evolves according to the following rule: for each site x ∈ Λ, containing ηx particles, with probability rate c(ηx), one particle jumps from x to one of

We observe that a class of conditional probability measures for unbounded spin systems with convex interactions satisfies Poincaré and logarithmic Sobolev inequalities.

We construct two types of equilibrium dynamics of infinite particle systems in a Riemannian manifold X. These dynamics are analogs of the Glauber, respectively Kawasaki dynamics of lattice spin systems. The Glauber dynamics now is a process where interacting particles randomly appear and disappear, i.e., it is a birth-and-death process in X, while in the Kawasaki dynamics interacting particles randomly jump over X. We establish conditions on a priori explicitly given symmetrizing measures and generators of both dynamics under which corresponding conservative Markov processes exist.

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 Kawasaki-type dynamics in the lattice without exclusion, and in the continuum. New estimates for Glauber-type dynamics are also obtained. 1 1

We consider the anisotropic three dimensional XXZ Heisenberg ferromagnet in a cylinder with axis along the 111 direction and boundary conditions that induce ground states describing an interface orthogonal to the cylinder axis. Let L be the linear size of the basis of the cylinder. Because of the breaking of the continuous symmetry around the ^ z axis, the Goldstone theorem implies that the spectral gap above such ground states must tend to zero as L !1. In [3] it was proved that, by perturbing in a sub{cylinder with basis of linear size R L the interface ground state, it is possible to construct excited states whose energy gap shrinks as R . Here we prove that, uniformly in the height of the cylinder and in the location of the interface, the energy gap above the interface ground state is bounded from below by const.L . We prove the result by rst mapping the problem into an asymmetric simple exclusion process on Z then by adapting to the latter the recursive analysis to estimate from below the spectral gap of the associated Markov generator developed in [7]. Along the way we improve some bounds on the equivalence of ensembles already discussed in [3] and we establish an upper bound on the density of states close to the bottom of the spectrum. 2000 MSC: 82B10, 82B20, 60K35 Key words and phrases: XXZ model, quantum interface, asymmetric exclusion process, equivalence of ensembles, spectral gap. Date: June 25, 2001. 1.

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Spectral gap estimates for interacting particle systems via a Bochner–type identity

Spectral gap estimates for interacting particle systems via a Bochner–type identity