Acknowledgements

I am indebted to my advisor, Dr. Ofer Zeitouni, for his guidance, encouragement and support. Especially, he introduced to me the model of branching random walks and offered a lot of enlightening discussion on different topics in this thesis; under his en-couragement and support, I went to many probability conferences which boosted my research and broadened my horizon; he generously supported my research with his funding NSF grant DMS-0804133. I am grateful to my committee chair, Dr. Maury Bramson. He provided related references to several projects in this thesis; he proofread this thesis very carefully and explained in detail how to improve the mathematical writing. I would also like to thank the other two committee members: Dr. John Baxter and Dr. Bert Fristedt. Thank you for taking time out of your busy schedule to help me with the thesis. My gratitude also extends, but does not limit, to Dr. Nicolai V. Krylov, Dr. Arn