Phase Transitions in Random Interacting Systems

本研究项目涉及离散概率中的几个问题。第一个是关于随机相互作用的粒子系统，它可以作为从感染传播到竞争生物种群增长等现象的概率模型。即使是这种粒子系统中最简单的例子也没有被完全理解。这个项目的重点是一个系统在一个更大的家族被称为a + B - 2B模型被物理学家研究为燃烧模型。该系统最近在有限树和无限树上显示出令人惊讶的行为，该项目将进一步探索。另一个需要研究的主要问题是随机树，这是整个科学领域的另一个常见模型。本项目研究由随机树产生的方程，其解与树的某些分类是否存在相关。该项目的第一部分是研究一个相互作用的随机漫步系统，称为青蛙模型。在这个过程中，不活跃的粒子被放置在一个图上。然后一个粒子变得活跃并进行随机游走，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，以此类推。PI最近的工作表明，该模型在无限树上表现为瞬态和递归之间的相变，在有限树上表现为不同覆盖时间制度之间的相变。该项目提出了在无限树上存在一个附加的弱递归相位和在有限树上存在一个中间覆盖时间相位的证明。实现这一目标的一个潜在途径在于对递归分布方程进行更细致的分析，首先使用玩具模型。另一个提议的项目是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对树进行分类。这种分类遵循一个递归规则（即当且仅当至少有两个根子树在类中），该规则可以由树自动机描述。这个规则引出了一个定点方程，人们可以用它来计算高尔顿-沃森树在这类中的概率。本课题将进一步探讨分类、树自动机和不动点方程之间的关系。研究该问题的两种方法是随机树及其分支过程的概率研究和不动点方程的直接解析研究。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。

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.