Estimate of spectral gap for continuous gas

Abstract

Consider the continuous gas in a bounded domain Λ of R d , described by a Gibbsian measure μ Λ η associated with a pair interaction ϕ, the inverse temperature β, the activity z>0, and the boundary condition η. When ϕ is nonnegative, we show that the spectral gap of a Glauber type dynamic (i.e., some Markov process reversible with respect to μ Λ η ) in L 2( μ Λ η ) is bounded from below by 1−z∫ R d |1−e −βϕ(y)| dy and from above by 1+z∫ R d |1−e −βϕ(y)| dy , independent of Λ and η. This result improves a previous work by L. Bertini et al. (2002) and is extended also to the hard core case. Our approach consists to approximate the continuous gas model by the discrete spin model and to apply the M- ε theorem of Ligget. Some other results such as uniqueness, exponential convergence of the Glauber dynamic w.r.t. norms of Ligget's type are also obtained.