Phase Transitions in Random Interacting Systems

本研究项目解决了离散概率中的几个问题。第一个是关于粒子的随机相互作用系统，它作为一系列现象的概率模型，从感染的传播到竞争有机体的种群增长。即使是这样的粒子系统的最简单的例子也没有被完全理解。这个项目的重点是物理学家作为燃烧模型研究的一个更大的家族中的一个系统，即A+B-2B模型。该系统最近被证明在有限和无限树上显示出令人惊讶的行为，本项目将进一步探索这一点。另一个需要解决的主要问题是随机树，这是科学界另一个常见的模型。这个项目研究了由随机树产生的方程，其解与树的某些分类的存在或不存在有关。该项目的第一部分是研究一种称为青蛙模型的相互作用的随机游动系统。在这个过程中，不活跃的粒子被放置在图形上。然后，一个粒子变为活动状态并执行随机行走，唤醒它遇到的任何粒子。然后，这些粒子开始它们自己的随机行走，唤醒它们遇到的任何粒子，依此类推。PI最近的工作表明，该模型在无限树上的瞬变和递归之间以及在有限树上的不同覆盖时间区域之间呈现出相变。证明了无限树上存在一个附加的弱递归阶段，有限树上存在一个中间覆盖时间阶段。一个潜在的途径是对递归分布方程进行进一步的更精细的分析，首先研究玩具模型。另一个被提议的项目探索遵循自动机给出的规则的树的分类。典型的例子是根据树是否包含从根开始的无限二叉子树来对树进行分类。这种分类遵循可以由树自动机描述的递归规则(即，当且仅当至少两个根子子树在类中时，树才在类中)。这一规则导出了一个不动点方程，可以用来计算Galton-Watson树属于这类树的概率。这个项目将进一步研究分类、树自动机和不动点方程之间的关系。解决这个问题的两种方法是对随机树和由此产生的其他分支过程的概率调查，以及对不动点方程的直接分析调查。这一奖项反映了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。

This research project addresses several questions in discrete probability. The first is on random interacting systems of particles, which serve as probabilistic models for phenomena ranging from the spread of infections to the growth of populations of competing organisms. Even the simplest examples of such particle systems are not fully understood. This project is focused on a system in a larger family known as A + B - 2B models studied by physicists as models for combustion. The system has recently been shown to display surprising behavior on finite and infinite trees, which this project will explore further. The other major question to be pursued is on random trees, another common model throughout the sciences. This project investigates equations arising from random trees whose solutions are related to the existence or nonexistence of certain classifications of trees into categories.The first part of the project is to study a system of interacting random walks known as the frog model. In this process, inactive particles are placed on a graph. One particle then becomes active and performs a random walk, waking any particles it encounters. These particles then start their own random walks, waking any particles they encounter, and so on. Recent work by the PI has shown that the model exhibits phase transitions between transience and recurrence on infinite trees and between different cover time regimes on finite trees. The project proposes proving the existence of an additional weak recurrence phase on the infinite tree and an intermediate cover time phase on the finite tree. A potential route to this lies in further finer analysis of recursive distributional equations, working first with toy models. The other proposed project explores classifications of trees that follow rules given by automata. The prototypical example is the classification of trees according to whether they contain an infinite binary subtree starting at the root. This classification obeys a recursive rule (namely that a tree is in the class if and only if at least two root child subtrees are) that can be described by a tree automaton. This rule induces a fixed-point equation that one can use to compute the probability of a Galton-Watson tree being in this class. This project will investigate further the relationship between the classifications, the tree automata, and the fixed-point equations. Two approaches to the problem are probabilistic investigation of the random trees and other branching processes arising from them, and the direct analytic investigation of the fixed-point 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.