TOPICS IN RANDOM WALKS by AARON MONTGOMERY A DISSERTATION Presented to the

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TOPICS IN RANDOM WALKS
by
AARON MONTGOMERY
A DISSERTATION
Presented to the Department of Mathematics
and the Graduate School of the University of Oregon
in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
June 2013
DISSERTATION APPROVAL PAGE
Student: Aaron Montgomery
Title: Topics in Random Walks
This dissertation has been accepted and approved in partial fulfillment of the
requirements for the Doctor of Philosophy degree in the Department of Mathematics
by:
David Levin Chair
N. Christopher Phillips Member
Chris Sinclair Member
Yuan Xu Member
Reza Rejaie Outside Member
and
Kimberly Andrews Espy Vice President for Research & Innovation/
Dean of the Graduate School
Original approval signatures are on file with the University of Oregon Graduate
School.
Degree awarded June 2013
ii
(cid:13)c 2013 Aaron Montgomery
iii
DISSERTATION ABSTRACT
Aaron Montgomery
Doctor of Philosophy
Department of Mathematics
June 2013
Title: Topics in Random Walks
We study a family of random walks defined on certain Euclidean lattices that are
related to incidence matrices of balanced incomplete block designs. We estimate the
return probability of these random walks and use it to determine the asymptotics of
the number of balanced incomplete block design matrices. We also consider the
problem of collisions of independent simple random walks on graphs. We prove
some new results in the collision problem, improve some existing ones, and provide
counterexamples to illustrate the complexity of the problem.
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CURRICULUM VITAE
NAME OF AUTHOR: Aaron Montgomery
GRADUATE AND UNDERGRADUATE SCHOOLS ATTENDED:
University of Oregon, Eugene, OR
Northwest Nazarene University, Nampa, ID
DEGREES AWARDED:
Doctor of Philosophy, Mathematics, 2013, University of Oregon
Master of Science, Mathematics, 2009, University of Oregon
Bachelor of Science, Physics, 2006, Northwest Nazarene University
Bachelor of Science, Mathematics, 2006, Northwest Nazarene University
AREAS OF SPECIAL INTEREST:
Stochastic processes, Markov chains and their applications, potential theory,
combinatorial design theory
PROFESSIONAL EXPERIENCE:
Graduate Teaching Fellow, University of Oregon, 2006-2013
GRANTS, AWARDS AND HONORS:
Donald and Darel Stein Graduate Student Teaching Award, University of
Oregon, 2011
Frank W. Anderson Graduate Teaching Award, University of Oregon
Mathematics Department, 2011
Dan Kimble First Year Teaching Award, University of Oregon, 2007
Summa cum Laude, Northwest Nazarene University, 2006
v
ACKNOWLEDGEMENTS
I would like to thank my advisor, Professor David Levin, for providing guidance
and direction. I would also like to thank Professor Chris Phillips for his thorough
comments during the revision process. I would also like to acknowledge the work of
the late Warwick de Launey, whose notes provided useful insight in the beginning
stages of my work.
vi
For my wife, Melissa.
vii
TABLE OF CONTENTS
Chapter Page
I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
II. COUNTING BALANCED INCOMPLETE BLOCK DESIGN
INCIDENCE MATRICES . . . . . . . . . . . . . . . . . . . . . . . 5
2.1. Extreme Values of the Characteristic Function . . . . . . . . . 11
2.2. Anatomy of the Integral . . . . . . . . . . . . . . . . . . . . . 20
2.3. Bounds Far from the Maximal Set . . . . . . . . . . . . . . . . 30
2.4. Bounds Near the Maximal Set . . . . . . . . . . . . . . . . . . 35
2.5. The Submatrix Determinant . . . . . . . . . . . . . . . . . . . 45
2.6. Proof of Main Theorem . . . . . . . . . . . . . . . . . . . . . . 64
2.7. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2.8. Supplementary Material . . . . . . . . . . . . . . . . . . . . . 75
III. COLLISIONS OF INDEPENDENT RANDOM WALKS ON
GRAPHS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.1. Counterexamples in Collision Theory . . . . . . . . . . . . . . 83
3.2. Truncations of the Comb Graph . . . . . . . . . . . . . . . . . 92
3.3. Stability of the Green’s Function Criterion . . . . . . . . . . . 99
3.4. Quasi-transitive Graphs . . . . . . . . . . . . . . . . . . . . . . 114
3.5. Quadruple-collisions . . . . . . . . . . . . . . . . . . . . . . . . 121
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Chapter Page
3.6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
REFERENCES CITED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
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LIST OF FIGURES
Figure Page
3.1. The graph G. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.2. The graph H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
1
3.3. The graph H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
2
3.4. A certain quasi-transitive truncation of Comb(Z,Z). . . . . . . . . . . . 119
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