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，如伊辛模型和渗流模型，这些模型对他们所建模的物理系统的相变提供了重要的见解。随机空间模型是重要的应用，因此，重要的是要发展的方法来理解他们的定性行为。对于许多系统，如接触过程，这是一个简单的模型的传播和竞争的物种，非平凡的平稳分布是唯一的，它的密度是一个连续的函数的参数。在这个项目中进行的研究将调查一些系统，这条规则的例外。PI在非线性投票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态的例子。PI最近表明，一个易感染的流行病，其中个人下降连接到受感染的个人可以有一个不连续的相变。目标之一是将其扩展到更现实的SIR模型。在另一个方向上，有一个称为爆炸渗流的过程，该过程在2009年基于模拟被证明具有不连续的相变，但一年后，一个严格的数学证明表明该系统和许多类似的系统具有连续的相变。该项目的目标之一是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。工作将进行四个具有挑战性和长期研究的问题的概率。为了简洁起见，这里只提到三个：（i）爆炸渗流发生在一个动态生长的随机图上，其中在每一步中选择m个潜在的新边，但只选择添加。Achlioptas，D'Souza和Spencer在2009年声称这些模型可以有不连续的过渡，但一年后Riordan和Warnke表明，对于一大类规则，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的信息。(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。最近黄和Durrett表明，收敛到反应扩散方程的几个粒子系统的零波速收敛到运动的平均曲率随着时间的进一步加快。这为证明不连续相变的存在提供了可能性。(iii)Durrett和Yao最近在演化组态模型图上给出了SI模型不连续相变的一个几乎充要条件。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存在的悬而未决的问题。这个奖项反映了NSF的法定使命，并已被认为是值得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的支持。

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.