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.
Highlights
The process studied in this paper is an analogue for continuous systems of the wellknown Glauber dynamics for lattice systems
The infinitesimal generator L associated to these dynamics have a spectral gap for small positive potentials and small activity
We introduce a class of non-trivial potentials for which the spectral gap has at least the same size as in the free case and, even more surprisingly, the derived bound on the size of the spectral gap is independent of the activity
Summary
The process studied in this paper is an analogue for continuous systems of the wellknown Glauber dynamics for lattice systems. The infinitesimal generator L associated to these dynamics have a spectral gap for small positive potentials and small activity (high temperature regime). We introduce a class of non-trivial potentials for which the spectral gap has at least the same size as in the free case and, even more surprisingly, the derived bound on the size of the spectral gap is independent of the activity. The definition of this class is based upon Fourier transform and the continuous space structure of the system is essential. These potentials have a non-trivial attractive part, there will be no phase transition of any kind for all values of the activity z
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have