Multitype Particle Systems

具有随机动力学的相互作用的粒子系统是物理和社会科学中模拟现象的基础。这样的系统可以用来描述化学反应，以及通过网络传播疾病、信息和物种。当合并多个粒子类型时，这些模型通常会变得更有意义。例如，著名的第一通道渗流模型描述了单一物种在环境中的传播；竞争物种的加入丰富了该模型。这个项目寻求研究化学反应、流行病爆发和信息传播的著名模型的更现实的变体，以加深我们对科学上的重要现象的理解，并进一步发展有助于解释它们的数学。该项目将涉及本科生的培训。本项目将考虑五个不同的随机多类型相互作用粒子系统：(1)扩散-有限湮灭模型是一个湮灭粒子系统，其中相对类型的粒子之间的碰撞导致相互湮灭。粒子具有不同速度的版本很难研究，并推测当不同类型的粒子最初达到平衡时，会显示出异常行为。(2)弹道湮灭模型有三种粒子类型，所有碰撞都会导致湮灭。对不同区域的极限颗粒密度进行了表征。(3)Chase-Escape模型是一个竞争性随机增长模型，当捕食者比被捕食者适合度高时，假设存在共存阶段。将调查环境和不同物种的适合度对共存的影响。(4)A+B-2A模型描述了感染颗粒的生长系统。这一过程最近被推广到连续体，这导致了离散概率和连续概率相交的几个新问题，这些问题将被解决。(5)在临界连通门限附近的随机图上，经典的易感感染移除模型被很好地理解。一个自然的延伸是让易感人群重新连接他们与感染者的联系。这一更现实的修改可能会改变流行病的规模。这五个过程将使用离散和连续概率理论的各种工具进行研究，如质量传输原理、递归、耦合、大偏差估计和随机微分方程。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Interacting particle systems with random dynamics are fundamental for modeling phenomena in the physical and social sciences. Such systems can be used to describe chemical reactions, as well as the spread of disease, information, and species through a network. These models often become more meaningful when multiple particle types are incorporated. For example, the celebrated First Passage Percolation model describes the spread of a single species through an environment; the incorporation of competing species enriches the model. This project seeks to study more realistic variants of well-known models for chemical reactions, epidemic outbreaks, and the spread of information as to deepen our understanding of important phenomena from across the sciences and further develop the mathematics that helps explain them. The project will involve the undergraduate students training.This project will consider five different stochastic multi-type interacting particle systems: (1) The Diffusion-Limited Annihilation model is an annihilating particle system in which collisions between opposite type particles result in mutual annihilation. The version in which particles have different speeds is difficult to study and conjectured to exhibit anomalous behavior when different particle types are initially balanced. (2) The Ballistic Annihilation model has three particle types and all collisions result in annihilation. The limiting particle density in different regimes will be characterized. (3) The Chase-Escape model is a competitive stochastic growth model conjectured to have a coexistence phase even when the predatory species is fitter than the prey. The effect of the environment and fitness of the different species on coexistence will be investigated. (4) The A+B-2A model describes a growing system of infected particles. This process was recently generalized to the continuum, which leads to several new questions at the intersection of discrete and continuous probability, which will be addressed. (5) Lastly, the classical Susceptible-Infected-Removed model is well-understood on random graphs near the critical connectivity threshold. A natural extension is to have susceptible individuals rewire their connections away from infected individuals. This more realistic modification may change the size of epidemics. These five processes will be studied using a variety of tools from discrete and continuous probability theory such as the mass transport principal, recursion, couplings, large deviation estimates, and stochastic differential equations.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.