Interacting Particle Systems on Lattices and on Graphs

该项目关注由各种应用驱动的生态和社会互动的空间模型。这项研究的主题是观察当先前在假设每个个体与所有其他个体相互作用的情况下研究的系统通过纳入空间而变得更加现实时，预测是如何变化的。四个主要的例子如下：(i) Staver-Levin森林模型，该模型预测森林和稀树草原（有孤立树木的草地）是交替的稳定状态；（ii）进化博弈，长期以来在生态学中被用来解释利他行为的持续性等现象；（iii）阿克塞尔罗德模型，该模型研究当个人与基于他们分享的意见数量的概率互动时意见的传播；(4)潜在选民模型（latent voter model），研究技术在社会网络中的传播，当消费者刚刚获得新产品时，他们会等待一段时间才愿意购买新产品。研究这些理想模型的总体目标是了解系统的平衡特性如何依赖于相互作用的细节。当每个个体与所有其他个体相互作用时，系统是一个常微分方程，很容易研究。然而，当明确地考虑到空间时，问题就变得非常困难。本项目有以下具体目标：(i)表明在Staver-Levin模型中，森林和草原之间边界的移动方向表明了空间模型中真正平衡的一种状态；（ii）表明进化博弈具有三种独立的弱选择机制，可以导致PDE、ODE或Tarnita公式有效的机制；（ii）在存在大量问题且存在大量意见的情况下，完成李俊驰研究阿克塞尔罗德模型的论文工作（这将为该模型在二维上提供第一个严格的结果）；（iv）表明，即使潜伏期很短，它也会改变动态，因此只有一个非平凡平稳分布，而没有潜伏期的选民模型只有一个参数族。

This project concerns spatial models for ecological and social interactions motivated by various applications. The theme of this research is to see how predictions change when systems previously studied under the assumption that each individual interacts with all the others are made more realistic by incorporating space. The four main examples are the following: (i) the Staver-Levin forest model, which predicts that forest and savannah (grassland with isolated trees) are alternative stable states; (ii) evolutionary games, which have long been used in ecology to explain phenomena such as the persistence of altruistic behavior; (iii) Axelrod's model, which studies the spread of opinions when individuals interact with a probability based on the number of the number of opinions they share; (iv) the latent voter model, which studies the spread of technology in a social network when consumers who have just acquired a new product will wait some time before they are willing to purchase a new one. The general goal of studying these idealized models is to understand how properties of the equilibrium of the system depend on the details of the interactions. When each individual interacts with all the others, the system is an ordinary differential equation and is easily studied. However, when space is explicitly taken into account the problems become very difficult. This project has the following specific goals: (i) show that in the Staver-Levin model, the direction of movement of a boundary between forest and savannah indicates the one state that is the true equilibrium in the spatial model; (ii) show that evolutionary games have three separate weak selection regimes that can lead to a PDE, ODE, or a regime in which Tarnita's formulas are valid; (ii) complete Junchi Li's thesis work studying Axelrod's model in the situation in which there are a large number of issues about which there are a large number of opinions (this would provide the first rigorous result for that model in two dimensions); (iv) show that even if latent period is brief, it changes the dynamics so that there is only one nontrivial stationary distribution, in contrast to the one parameter family in the voter model without latency.