标题： 一种具有对数线性相互作用的相互作用自旋/队列模型的扩散极限 显示英文标题

标题： A diffusion limit for a model of interacting spins/queues with log-linear interaction

摘要： 在本文中，我们建立了关于一个相互作用自旋模型的扩散极限，该模型定义为多组分马尔可夫链，其组分（自旋）由有限图的顶点索引。自旋取值于非负整数的有限集合，并在基于图的对数线性相互作用下演化。我们证明，如果可能的自旋值集合扩展到所有非负整数，则马尔可夫链的时间缩放和归一化版本收敛到一个在原点反射的相互作用奥恩斯坦-乌伦贝克过程系统。这种极限类似于排队论中的重载极限（并且我们的模型可以自然地解释为排队模型）。我们的证明借鉴了排队论的发展，并依赖于鞅方法。 显示英文摘要

摘要： In this paper we establish a diffusion limit for an interacting spin model defined in terms of a multi-component Markov chain whose components (spins) are indexed by vertices of a finite graph. The spins take values in a finite set of non-negative integers and evolve subject to a graph based log-linear interaction. We show that if the set of possible spin values expands to the set of all non-negative integers, then a time-scaled and normalised version of the Markov chain converges to a system of interacting Ornstein-Uhlenbeck processes reflected at the origin. This limit is akin to heavy traffic limits in queueing (and our model can be naturally interpreted as a queueing model). Our proof draws on developments in queueing theory and relies on martingale methods.