Seminars｜Fostering top leaders in mathematics -- broadening the core and exploring new ground

Seminar

Language	English
From	2013/08/02 15:30
To	2013/08/02 17:00
Place	Room 552, Building No.3, Faculty of Science, Kyoto University
Seminar Name	Kansai Probability Theory Seminar
Title	Growth exponent for loop-erased random walk in three dimensions
Field	Analysis
Speakers	Daisuke Shiraishi
Affiliation	Kyoto
Abstract	Let $M_{n}$ be the expected number of steps of the loop-erasure of a simple random walk on $\mathbb{Z}^{3}$ run until its first exit from a ball of radius $n$ . It is believed that there exists $\alpha_{3}$ (called growth exponent) such that the following holds; $\lim_{n \to <br>\infty} M_{n}/ n^{\alpha} = 0$ for all $\alpha > \alpha_{3}$ , and $\lim_{n \to \infty} M_{n}/ n^{\alpha} = \infty$ for all $\alpha < <br>\alpha_{3}$ . However, rigorously the existence of $\alpha_{3}$ is not proved. In the talk, I will show the existence of the growth exponent $\alpha_{3}$ in the weak sense, i.e., $\lim_{n \to \infty} M_{n}/ <br>n^{\alpha} = 0$ for all $\alpha > \alpha_{3}$ , and $\limsup_{n \to <br>\infty} M_{n}/ n^{\alpha} = \infty$ for all $\alpha < \alpha_{3}$ . I will explain how to prove the existence of $\alpha_{3}$ via the non-intersecting random walk.
Link	http://www.math.kyoto-u.ac.jp/probability/seminar/index_j.html