Gibbs点过程几何泛函的稳定性和极限定理
Stabilization and limit theorems for geometric functionals of Gibbs point processes
摘要：
给定$\R^d$上的Gibbs点过程$¶^Ψ$具有足够弱的势$Ψ$，我们考虑随机测度$μ_\la := \sum_{x \in ¶^Ψ \cap Q_\la} ξ(x, ¶^Ψ \cap Q_\la) δ_{x/\la^{1/d}}$，其中$Q_{\la} := [-\la^{1/d}/2,\la^{1/d}/2]^d$是体积$\la$立方体，$ξ(\cdot,\cdot)$是平移不变稳定泛函.在$Ψ$满足局部化性质和平移不变性的条件下，建立了$\la^{-1} μ_\la(f)$的弱大数定律，$f$是$\R^d$上的有界测试函数，$\la^{-1/2} μ_\la(f),$弱收敛于作用于有界测试函数的高斯场.结果产生的吉布斯点过程，包括施特劳斯和面积相互作用点过程，以及更一般的点过程定义的Widom-Rowlinson和硬核模型的几何泛函的极限定律。我们提供的应用程序随机序列包装的Gibbsian输入，欧几里德图，网络和渗流模型的泛函的Gibbsian输入，并通过Gibbsian输入量化。
英文摘要：
Given a Gibbs point process $¶^Ψ$ on $\R^d$ having a weak enough potential $Ψ$, we consider the random measures $μ_\la := \sum_{x \in ¶^Ψ \cap Q_\la} ξ(x, ¶^Ψ \cap Q_\la) δ_{x/\la^{1/d}}$, where $Q_{\la} := [-\la^{1/d}/2,\la^{1/d}/2]^d$ is the volume $\la$ cube and where $ξ(\cdot,\cdot)$ is a translation invariant stabilizing functional. Subject to $Ψ$ satisfying a localization property and translation invariance, we establish weak laws of large numbers for $\la^{-1} μ_\la(f)$, $f$ a bounded test function on $\R^d$, and weak convergence of $\la^{-1/2} μ_\la(f),$ suitably centered, to a Gaussian field acting on bounded test functions. The result yields limit laws for geometric functionals on Gibbs point processes including the Strauss and area interaction point processes as well as more general point processes defined by the Widom-Rowlinson and hard-core model. We provide applications to random sequential packing on Gibbsian input, to functionals of Euclidean graphs, networks, and percolation models on Gibbsian input, and to quantization via Gibbsian input.
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