Electronic Journal of Probability

Spectral gap for Glauber type dynamics for a special class of potentials

Yuri Kondratiev, Tobias Kuna, and Natascha Ohlerich

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We consider an equilibrium birth and death type process for a particle system in infinite volume, the latter is described by the space of all locally finite point configurations on $\mathbb{R}^{d}$. These Glauber type dynamics are Markov processes constructed for pre-given reversible measures. A representation for the "carré du champ" and "second carré du champ" for the associate infinitesimal generators $L$ are calculated in infinite volume and for a large class of functions in a generalized sense. The corresponding coercivity identity is derived and explicit sufficient conditions for the appearance and bounds for the size of the spectral gap of $L$ are given. These techniques are applied to Glauber dynamics associated to Gibbs measure and conditions are derived extending all previous known results and, in particular, potentials with negative parts can now be treated. The high temperature regime is extended essentially and potentials with non-trivial negative part can be included. Furthermore, a special class of potentials is defined for which the size of the spectral gap is as least as large as for the free system and, surprisingly, the spectral gap is independent of the activity. This type of potentials should not show any phase transition for a given temperature at any activity.

Article information

Electron. J. Probab., Volume 18 (2013), paper no. 42, 18 pp.

Accepted: 23 March 2013
First available in Project Euclid: 4 June 2016

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82C21: Dynamic continuum models (systems of particles, etc.) 82C22: Interacting particle systems [See also 60K35] 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 58J50: Spectral problems; spectral geometry; scattering theory [See also 35Pxx]

Birth-and-death process continuous system Glauber dynamics spectral gap absence of phase transition

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Kondratiev, Yuri; Kuna, Tobias; Ohlerich, Natascha. Spectral gap for Glauber type dynamics for a special class of potentials. Electron. J. Probab. 18 (2013), paper no. 42, 18 pp. doi:10.1214/EJP.v18-2260. https://projecteuclid.org/euclid.ejp/1465064267

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