Existence of Gibbsian point processes with geometry-dependent interactions

Regular $g$-measures are not always Gibbsian

Regular $g$-measures are discrete-time processes determined by conditional expectations with respect to the past. One-dimensional Gibbs measures, on the other hand, are fields determined by simultaneous conditioning on past and future. For the Markovian and exponentially continuous cases both theories are known to be equivalent. Its equivalence for more general cases was an open problem. We present a simple example settling this issue in a negative way: there exist $g$-measures that are continuous and non-null but are not Gibbsian. Our example belongs, in fact, to a well-studied family of proc…

Gibbs States and Gibbsian Specifications on the space $\mathbb{R}^{\mathbb{N}}$

We are interested in the study of Gibbs and equilbrium probabilities on the lattice $\mathbb{R}^{\mathbb{N}}$. Consider the unilateral full-shift defined on the non-compact set $\mathbb{R}^{\mathbb{N}}$ and an $α$-Hölder continuous potential $A$ from $\mathbb{R}^{\mathbb{N}}$ into $\mathbb{R}$. From a suitable class of a priori probability measures $ν$ (over the Borelian sets of $\mathbb{R}$) we define the Ruelle operator associated to $A$ (using an adequate extension of this operator to the compact set $\overline{\mathbb{R}}^\mathbb{N}=(S^1)^\mathbb{N}$) and we show the existence of eigenfunc…

抽象空间中Gibbs点过程的结构性质

Structural properties of Gibbsian point processes in abstract spaces

本文在任意可测空间中以GNZ方程为定义研究Gibbs过程的结构性质，推广了配分函数、Janossy密度和Hamilton量等概念，证明了局部收敛子序列提取定理，并由此确立了具有无限相互作用范围及团簇依赖相互作用的Gibbs过程与粒子过程的存在性。

Gibbsian representation for point processes via hyperedge potentials

We consider marked point processes on the d-dimensional euclidean space, defined in terms of a quasilocal specification based on marked Poisson point processes. We investigate the possibility of constructing absolutely-summable Hamiltonians in terms of hyperedge potentials in the sense of Georgii et al. These potentials are a natural generalization of physical multi-body potentials which are useful in models of stochastic geometry. We prove that such representations can be achieved, under appropriate locality conditions of the specification. As an illustration we also provide such potential re…

强相关点场中的近似吉布斯结构与广义高斯零点系综

Approximate Gibbsian structure in strongly correlated point fields and generalized Gaussian zero ensembles

本文为强相关随机点场提出近似吉布斯结构的一般框架，并证明广义吉布斯性质对任意 $α$ 值的 $α$-GAFs 成立，从而确定其零点过程的刚性水平恰好为 $\lfloor \frac{1}{α} \rfloor$，解决了任意刚性水平点过程存在性的开放问题。

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

We present general existence and uniqueness results for marked models with pair interactions, exemplified through Gibbs point processes on path space. More precisely, we study a class of infinite-dimensional diffusions under Gibbsian interactions, in the context of marked point configurations: the starting points belong to $\mathbb{R}^d$, and the marks are the paths of Langevin diffusions. We use the entropy method to prove existence of an infinite-volume Gibbs point process and use cluster expansion tools to provide an explicit activity domain in which uniqueness holds.