Four Challenging Questions in Probability

数学家和物理学家长期以来一直在研究随机空间模型，比如伊辛模型和渗透，它们为他们所建模的物理系统的相变提供了重要的见解。随机空间模型在应用中很重要，因此开发理解其定性行为的方法非常重要。对于许多系统，如接触过程，它是物种传播和竞争的简单模型，其非平凡平稳分布是唯一的，其密度是其参数的连续函数。在这个项目中进行的研究将调查一些例外的系统。PI在非线性选民模型上的工作旨在产生可以严格证明具有两个非平凡平移不变平衡状态的例子。PI最近表明，个体与受感染个体断绝联系的易感感染流行病可能具有不连续的相变。目标之一是将其扩展到更现实的SIR模型。在另一个方向上，有一种叫做爆炸渗流的过程，在2009年基于模拟推测它具有不连续的相变，但一年后一个严格的数学证明表明，这个系统和许多类似的系统具有连续的相变。该项目的目标之一将寻求证明爆炸渗流模型中相变的定性结果。该项目将为研究生提供研究培训机会。工作将在概率中四个具有挑战性和长期研究的问题上进行。为简洁起见，这里只提到三种情况：(i)爆炸渗透发生在动态增长的随机图上，其中每一步选择m个潜在的新边，但只选择要添加的边。Achlioptas、D’souza和Spencer在2009年声称，这些模型可能具有不连续的过渡，但一年后，Riordan和Warnke表明，对于一类广泛的规则来说，过渡是连续的。该项目的目标之一是将他们的结果扩展到其他系统，并获得有关相变的更详细信息。（ii）在许多情况下，通过证明粒子系统收敛于具有正波速的反应扩散方程，证明了随机空间模型的平稳分布的存在性。最近，Huang和Durrett已经证明，随着时间的进一步加速，几个收敛于零波速反应扩散方程的粒子系统会收敛于平均曲率的运动。这为证明不连续相变的存在提供了可能性。（iii） Durrett和Yao最近在演化的组态模型图上给出了SI模型中不连续相变的几乎充分必要条件。该项目的目标之一是将这一结果扩展到更现实的SIR和SIS流行病。后一个问题是一个长期悬而未决的问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Mathematicians and physicists have long studied stochastic spatial models, like the Ising model and percolation that give important insights into the phase transitions of the physical systems that they model. Stochastic spatial models are important for applications, so it is important to develop methods to understand their qualitative behavior. For many systems like the contact process, which is a simple model of the spread and competition of species, the nontrivial stationary distribution is unique and its density is a continuous function of its parameters. The research carried out in this project will investigate some systems that are exceptions to this rule. The PI's work on nonlinear voter models seeks to produce examples that can be rigorously shown to have two nontrivial translation invariant equilibrium states. The PI has recently shown that a susceptible-infected epidemics in which individuals drop connections to infected individuals can have a discontinuous phase transition. One of the goals is to extend this to the more realistic SIR model. In the other direction there is a process called explosive percolation that was conjectured in 2009 based on simulation to have a discontinuous transition, but one year later a rigorous mathematical proof showed that this system and a number of similar systems have continuous phase transitions. One of the goals of this project will seek to prove qualitative results about the phase transition in the explosive percolation model. The project will provide research training opportunities for graduate students. Work will be carried out on four challenging and long-studied problems in probability. For brevity only three are mentioned here: (i) Explosive percolation occurs on a dynamically grown random graph in which m potential new edges are chosen on each step but only is chosen to be added. Achlioptas, D’Souza and Spencer claimed in 2009 that these models could have discontinuous transitions, but one year later Riordan and Warnke showed that for a broad class of rules the transition is continuous. One of the goals of this project is to extend their results to other systems and to obtain more detailed information about the phase transition. (ii) In many cases the existence of stationary distributions for a stochastic spatial model has been proved by showing that the particle systems converge to reaction diffusion equations with positive wave speed. Recently Huang and Durrett have shown that several particle systems that converge to reaction-diffusion equations with zero wave speeds converge to motion by mean curvature as time is further sped-up. This opens up the possibility of proving the existence of discontinuous phase transitions. (iii) Durrett and Yao have recently given an almost necessary and sufficient condition for a discontinuous phase transition in the SI model on an evolving configuration model graph. One of the aims of this project is to extend this result to more realistic SIR and SIS epidemics. The latter question is a long standing open problem.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.