# Project Overview

## Matrix analysis of interacting multi-species particle systems

Joe Chen (jpchen@colgate.edu)

Mathematics

### Abstract

Random walk on a connected graph is a classic problem in mathematics that can be studied by analyzing
properties of the graph Laplacian matrix L. Important questions include: What are the stationary solutions
(the null vectors of L)? Eigenvalues and eigenvectors of L? And solutions to the heat equation (the matrix
exponential exp(tL))?
In this project we will address the same questions, but replace the random walk model by a model
of interacting particles which are subject to nearest-neighbor interactions. In other words, as long as the
particles are not on adjacent vertices, the random walk rules apply; but if they are adjacent, then a set of
”exclusion” rules takes hold, such as: one particle may not jump to a vertex already occupied by another
particle. Also, particles of the same type are assumed to be indistinguishable.
To analyze this one-species exclusion model requires writing down a 2n-by-2n matrix, where n is the
number of vertices of the graph. How daunting! However, by adopting an orthonormal Fourier basis (that
is frequently used to study Boolean functions), one can show that the matrix is sparse (has lots of 0’s) and
possesses a special block structure. I have worked out all the details, and will teach you the essential results
during the first 2 weeks of the project.
Starting in Week 3, we will tackle the exclusion model involving two species (or ”colors”) of particles.
This leads to a 3n-by-3n matrix, and we will search for a ”nice” orthonormal Fourier basis which yields the
cleanest-looking block matrix. Once this matrix is found, we will proceed to find its eigenvalues, eigenvectors,
and stationary and dynamical solutions. Some of these questions can be answered on general connected
graphs, while others are best answered on d-dimensional lattice graphs or discrete tori (the 1-dimensional
graphs have attracted the most interest).
As we unravel the details of the model together, you will learn fascinating connections of this problem to
analysis, probability, abstract algebra, and combinatorics.
An essential part of this ”laboratory experience” is to train you how to write and communicate math
effectively, including typing math reports using TeX / Overleaf, and giving poster and oral presentations.

Dates: May 28, 2024 (starts the Tuesday after Memorial Day) through July 26, 2024, for a total of 9 weeks.

### Student Qualifications

This project requires extensive use of linear algebra and pencil-and-paper computations, so a certain level of mathematical maturity and attention to detail is needed to maximize your success. You are expected to have completed, by the end of the spring semester, Linear Algebra (MATH 214) and a proof-writing course (MATH 250, or a 300-level course which has MATH 250 as a prereq) with very good grades. Any additional math coursework at or above the 300 level is considered a plus. Programming experience, while not required, is welcome.

Please address all of the above in your application. When applicable, describe your previous experience working on an intellectually challenging project, how you overcame the challenges, and the lessons you learned from that project.

1~2 student

### Project Length

9 weeks

Applications open on 10/03/2023 and close on 02/28/2024