时变稀疏随机网络上相互作用粒子系统的热力学极限
Thermodynamic Limit of Interacting Particle Systems over Time-varying Sparse Random Networks
摘要：
我们对相互作用粒子系统的可观测宏观状态变量建立了一个泛函弱大数定律（例如，投票者和接触过程）在快速时变稀疏随机网络的相互作用。我们发现，随着代理$N$的数量越来越大，代理$\left(\overline{Y}_{k}^{N}(t)\right)$的比例在一定的状态$k$收敛分布-或者，更准确地说，弱相对于\emph{càdlàg}样本路径的空间上的统一拓扑结构-在任何紧凑的间隔$\left[0,T\right]$的常微分方程的解决方案。虽然极限过程是马尔可夫的，但预极限过程，即，归一化的宏观状态向量过程$\left(\mathbf{\overline{Y}}^{N}(t)\right)=\left(\overline{Y}_{1}^{N}(t),\ldots,\overline{Y}_{K}^{N}(t)\right)$是非马尔可夫的，因为它们与系统的\emph{high-dimensional}微观状态相关，这排除了直接应用标准参数来建立弱收敛。建立弱收敛的文件中开发的技术可能是独立的利益。
英文摘要：
We establish a functional weak law of large numbers for observable macroscopic state variables of interacting particle systems (e.g., voter and contact processes) over fast time-varying sparse random networks of interactions. We show that, as the number of agents $N$ grows large, the proportion of agents $\left(\overline{Y}_{k}^{N}(t)\right)$ at a certain state $k$ converges in distribution -- or, more precisely, weakly with respect to the uniform topology on the space of \emph{càdlàg} sample paths -- to the solution of an ordinary differential equation over any compact interval $\left[0,T\right]$. Although the limiting process is Markov, the prelimit processes, i.e., the normalized macrostate vector processes $\left(\mathbf{\overline{Y}}^{N}(t)\right)=\left(\overline{Y}_{1}^{N}(t),\ldots,\overline{Y}_{K}^{N}(t)\right)$, are non-Markov as they are tied to the \emph{high-dimensional} microscopic state of the system, which precludes the direct application of standard arguments for establishing weak convergence. The techniques developed in the paper for establishing weak convergence might be of independent interest.
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