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Zbl 1042.60073
Wu, Liming
Estimate of spectral gap for continuous gas. (English)
[J] Ann. Inst. Henri Poincaré, Probab. Stat. 40, No. 4, 387-409 (2004). ISSN 0246-0203

Summary: Consider the continuous gas in a bounded domain $\Lambda$ of $\bbfR^d$, described by a Gibbsian measure $\mu^\eta_\Lambda$ associated with a pair interaction $\varphi$, the inverse temperature $\beta$, the activity $z>0$, and the boundary condition $\eta$. When $\varphi$ is nonnegative, we show that the spectral gap of a Glauber type dynamic (i.e., some Markov process reversible with respect to $\mu^\eta_\Lambda)$ in $L^2 (\mu^\eta_\Lambda)$ is bounded from below by $1-z\int_{\bbfR^d} \vert 1-e^{-\beta\varphi (y)}\vert dy$ and from above by $1+z \int_{\bbfR^d} \vert 1-e^{-\beta \varphi(y)}\vert dy$, independent of $\Lambda$ and $\eta$. This result improves a previous work by {\it L. Bertini}, {\it N. Cancrini} and {\it F. Cesi} [ibid. 38, No. 1, 91--108 (2002; Zbl 0994.82054)] 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 $M$-$\varepsilon$ theorem of Liggett. Some other results such as uniqueness, exponential convergence of the Glauber dynamic w.r.t. norms of Liggett's type are also obtained.

MSC 2000:: *60K40 Physical appl. of random processes
60J75 Jump processes
60G57 Random measures
82B20 Lattice systems
82C20 Dynamic lattice systems

Keywords: Poincaré inequality; Gibbs measures; birth and death processes

Citations: Zbl 0994.82054

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