Phase Transitions in Random Interacting Systems

Cuny College Of Staten Island, Staten Island NY

Investigators

Abstract

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.