2020 | An explicit continuum Dobrushin uniqueness criterion for Gibbs point processes with non-negative pair potentials | Pierre Houdebert, Alexander ZassLink zum Preprint
An explicit continuum Dobrushin uniqueness criterion for Gibbs point processes with non-negative pair potentials
Autoren: Pierre Houdebert, Alexander Zass
We present a uniqueness result for Gibbs point processes with interactions that come from a non-negative pair potential; in particular, we provide an explicit uniqueness region in terms of activity z and inverse temperature β. The technique used relies on applying to the continuous setting the classical Dobrushin criterion. We also present a comparison to the two other uniqueness methods of cluster expansion and disagreement percolation, which can also be applied for this type of interactions.
2020 | A Gibbs point process of diffusions: existence and uniqueness | Alexander ZassZeitschrift: Lectures in Pure and Applied MathematicsVerlag: Potsdam University PressBuchtitel: Proceedings of the XI international conference Stochastic and Analytic Methods in Mathematical PhysicsSeiten: 13-22Band: 6Link zur Publikation
Link zum Preprint
A Gibbs point process of diffusions: existence and uniqueness
Autoren: Alexander Zass
In this work we consider a system of infinitely many interacting diffusions as a marked Gibbs point process. With this perspective, we show, for a large class of stable and regular interactions, existence and (conjecture) uniqueness of an infinite-volume Gibbs process. In order to prove existence we use the specific entropy as a tightness tool. For the uniqueness problem, we use cluster expansion to prove a Ruelle bound, and conjecture how this would lead to the uniqueness of the Gibbs process as solution of the Kirkwood-Salsburg equation.
Lectures in Pure and Applied Mathematics
Potsdam University Press
Proceedings of the XI international conference Stochastic and Analytic Methods in Mathematical Physics
2020 | Marked Gibbs point processes with unbounded interaction: an existence result | Sylvie Roelly, Alexander ZassZeitschrift: Journal of Statistical PhysicsSeiten: 972–996Band: 179Link zur Publikation
Link zum Preprint
Marked Gibbs point processes with unbounded interaction: an existence result
Autoren: Sylvie Roelly, Alexander Zass
We construct marked Gibbs point processes in Rd under quite general assumptions. Firstly, we allow for interaction functionals that may be unbounded and whose range is not assumed to be uniformly bounded. Indeed, our typical interaction admits an as finite but random range. Secondly, the random marks - attached to the locations in Rd - belong to a general normed space S. They are not bounded, but their law should admit a super-exponential moment. The approach used here relies on the so-called entropy method and large-deviation tools in order to prove tightness of a family of finite-volume Gibbs point processes. An application to infinite-dimensional interacting diffusions is also presented.
Journal of Statistical Physics