## Electronic Journal of Probability

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

#### Abstract

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

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

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

https://projecteuclid.org/euclid.ejp/1465064267

Digital Object Identifier
doi:10.1214/EJP.v18-2260

Mathematical Reviews number (MathSciNet)
MR3040552

Zentralblatt MATH identifier
1285.60093

Rights

#### Citation

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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