Phase Transitions in Random Interacting Systems

这个研究项目解决了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，这些粒子系统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最简单的例子也没有被完全理解。该项目的重点是一个更大的家庭称为A + B -2 B模型的物理学家研究的燃烧模型的系统。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。另一个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。本课题研究随机树的解与树的分类有关的方程。课题的第一部分是研究相互作用的随机游动系统，即青蛙模型。在这个过程中，不活跃的粒子被放置在图表上。然后，一个粒子变得活跃并执行随机行走，唤醒它遇到的任何粒子。这些粒子然后开始它们自己的随机行走，唤醒它们遇到的任何粒子，等等。PI最近的工作表明，该模型在无限树上表现出瞬态和递归之间的相变，在有限树上表现出不同覆盖时间之间的相变。该项目提出证明无限树上存在一个额外的弱递归阶段，有限树上存在一个中间覆盖时间阶段。一个潜在的途径在于对递归分布方程进行更精细的分析，首先使用玩具模型。另一个提议的项目探索了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树来对树进行分类。这种分类遵循一个递归规则（即一棵树在类中当且仅当至少有两个根子树），可以用树自动机来描述。这个规则导出了一个不动点方程，可以用来计算一个高尔顿-沃森树在这个类中的概率。本计画将进一步探讨分类、树自动机与不动点方程式之间的关系。两种方法来解决这个问题的随机树和其他分支过程中产生的概率调查，并直接分析调查的定点equations.This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智力价值和更广泛的影响审查标准进行评估的支持。

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.