Gibbs point processes on path space: existence, cluster expansion and uniqueness

Abstract

We study a class of in nite-dimensional di usions under Gibbsian interactions, in the context of marked point con gurations: the starting points belong to ℝ , and the marks are the paths of Langevin di usions. We use the entropy method to prove existence of an in nitevolume Gibbs point process and use cluster expansion tools to provide an explicit activity domain in which uniqueness holds.

Gibbs measures for the repulsive Bose gas

“…With regards to recent work, we highlight the very accessible paper by Dereudre [Der09], in which the author considered geometry-dependent interaction between points in the plane R 2 . Recently, there has been interest in studying the existence of Gibbs measures for point processes decorated with random diffusion, see [Zas21] and [RZ20].…”

“…We strongly believe that this approach has merits beyond the Bose gas model. Note that the papers proving the existence of Gibbs measures for marked random diffusions (see [Zas21] for example) usually use (super-)exponential integrability conditions for the diameter.…”

On the uniqueness of Gibbs distributions with a non-negative and subcritical pair potential

“…This method can be traced back to [34]. For some recent contributions we refer to [19], which might also serve as a good survey, and [42]. The uniqueness intensity region identified by this method is characterized by the contractivity of certain integral operators and does not seem to have an explicit probabilistic interpretation.…”