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Wong, Ting Kam Leonard. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 91-97) and index. / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 1.1 --- Problems of fractal analysis --- p.6 / Chapter 1.2 --- The boundary theory approach --- p.7 / Chapter 1.3 --- Summary of the thesis --- p.9 / Chapter 2 --- Martin boundary --- p.13 / Chapter 2.1 --- Markov chains and discrete potential theory --- p.13 / Chapter 2.2 --- Martin compactification --- p.18 / Chapter 2.3 --- Convergence to boundary and integral representations --- p.20 / Chapter 2.4 --- Dirichlet problem at infinity --- p.25 / Chapter 3 --- Hyperbolic boundary --- p.27 / Chapter 3.1 --- Random walks on infinite graphs --- p.27 / Chapter 3.2 --- Hyperbolic compactification --- p.31 / Chapter 3.3 --- Ancona's theorem --- p.33 / Chapter 3.4 --- Self-similar sets as hyperbolic boundaries --- p.34 / Chapter 3.5 --- Hyperbolic compactification of augmented rooted trees --- p.44 / Chapter 4 --- Simple random walk on Sierpinski graphs --- p.47 / Chapter 4.1 --- Hcuristic argument for d = 1 --- p.48 / Chapter 4.2 --- Symmetries and group invariance --- p.51 / Chapter 4.3 --- Reflection principle --- p.54 / Chapter 4.4 --- Self-similar identity and hitting distribution --- p.60 / Chapter 4.5 --- Remarks and Open Questions --- p.64 / Chapter 5 --- Induced Dirichlet forms on self-similar sets --- p.66 / Chapter 5.1 --- Basics of Dirichlet forms --- p.67 / Chapter 5.2 --- Motivation: the classical Douglas integral --- p.68 / Chapter 5.3 --- Graph energy and the induced forms --- p.69 / Chapter 5.4 --- Induced Dirichlet forms on self-similar sets --- p.74 / Chapter 5.5 --- A uniform tail estimate via coupling --- p.83 / Chapter 5.6 --- Remarks and open questions --- p.89 / Index of selected terms --- p.98

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This thesis investigates uniform mixing on Cayley graphs over Z_3^d. We apply Mullin's results on Hamming quotients, and characterize the 2(d+2)-regular connected Cayley graphs over Z_3^d that admit uniform mixing at time 2pi/9. We generalize Chan's construction on the Hamming scheme H(d,2) to the scheme H(d,3), and find some distance graphs of the Hamming graph H(d,3) that admit uniform mixing at time 2pi/3^k for any k≥2. To restrict the mixing time, we derive a sufficient and necessary condition for uniform mixing to occur on a Cayley graph over Z_3^d at a given time. Using this, we obtain three results. First, we give a lower bound of the valency of a Cayley graph over Z_3^d that could admit uniform mixing at some time. Next, we prove that no Hamming quotient H(d,3)/<1> admits uniform mixing at time earlier than 2pi/9. Finally, we explore the connected Cayley graphs over Z_3^3 with connected complements, and show that five complementary graphs admit uniform mixing with earliest mixing time 2pi/9.

Thesis--Wisconsin. / Vita. Includes bibliographical references (leaf 81).

The classical method of solving random walk problems involves using Markov chain theory. When the particular random walk of interest is written in matrix form using Markov chain theory, the problem must then be solved using a digital computer. To solve all but the most trivial random walk problems by hand would be extremely difficult and time consuming. Very large random walk problems may even prove difficult to solve on the smaller digital computers. This paper intends to demonstrate a method that may be used to solve large random walk problems in a quick and economical manner. This alternate method uses resistive analogues and has the added feature of extracting particular solutions without having to completely solve the problem as would be necessary using a digital computer. Many analogues of random walks may also be quickly amended to include other random walks with relative ease using this alternate method of solution. Because this method uses nothing more than a power supply, a DC voltmeter and a set of resistors, the analogue of a particular random walk problem may be left set-up without incurring any loss of time or money on a digital computer. Once the resistors are mounted in a permanent fashion, the random walk analogues may also be used as an effective demonstration of random walk probabilities in the classroom.

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Includes bibliographical references (p. 131-134).

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