论文标题
抽象空间中Gibbsian Point过程的结构特性
Structural properties of Gibbsian point processes in abstract spaces
论文作者
论文摘要
用随机计数的语言测量泊松过程的许多结构特性，可以在任意可测量的空间中研究。我们提供了类似的吉布斯过程论文。以GNZ方程为这些对象的定义，Gibbs过程可以在抽象空间中引入而没有任何拓扑结构。在这种一般环境中，研究了分区功能，janossy密度和相关功能。尽管该定义涵盖了有限和无限的吉布斯处理，但即使在抽象空间中，有限的情况也允许通过熟悉的系列扩展进行等效且更明确的表征。对任意可测量空间的阶乘措施的最新概括，其中计数措施不能作为狄拉克措施的总和，同样允许概括哈密顿人的概念。完全表征Gibbs过程的DLR方程，以及局部收敛拓扑的基本结果，也以完全的一般性制定。我们证明了一种关于从一系列点过程中提取局部收敛子序列的新定理，并使用此语句为具有潜在无限范围相互作用的一般空间中的Gibbs过程提供了存在结果。这些结果用于保证具有簇依赖性相互作用的Gibbs过程的存在，并证明了有关Gibbsian粒子过程存在的最新猜想。
In the language of random counting measures many structural properties of the Poisson process can be studied in arbitrary measurable spaces. We provide a similarly general treatise of Gibbs processes. With the GNZ equations as a definition of these objects, Gibbs processes can be introduced in abstract spaces without any topological structure. In this general setting, partition functions, Janossy densities, and correlation functions are studied. While the definition covers finite and infinite Gibbs processes alike, the finite case allows, even in abstract spaces, for an equivalent and more explicit characterization via a familiar series expansion. Recent generalizations of factorial measures to arbitrary measurable spaces, where counting measures cannot be written as sums of Dirac measures, likewise allow to generalize the concept of Hamiltonians. The DLR equations, which completely characterize a Gibbs process, as well as basic results for the local convergence topology are also formulated in full generality. We prove a new theorem on the extraction of locally convergent subsequences from a sequence of point processes and use this statement to provide existence results for Gibbs processes in general spaces with potentially infinite range of interaction. These results are used to guarantee the existence of Gibbs processes with cluster-dependent interactions and to prove a recent conjecture concerning the existence of Gibbsian particle processes.
