Grant-in-Aid for Scientific Research (C)
|Allocation Type||Single-year Grants |
General mathematics (including Probability theory/Statistical mathematics)
|Research Institution||Keio University (2007)|
University of Tsukuba (2005-2006)
MINAMI Nariyuki Keio University, School of Medicine, Professor (10183964)
NAKAMO Fumihiko Keio University, Faculty of Science, Associate Professor (10291246)
NOMASA Ueki Kyoto University, Graduate School of Human and Environmental Studiess, Associate Professor (80211069)
MAKINO Hironori Tokai University, School of Information Ttchnology and Electrnics, Associate professor (40338786)
NAGAO Taro Nagoya University, Graduate School of Mathematics, Associate Professor (10263196)
鈴木 由紀 慶應義塾大学, 医学部, 講師 (30286645)
磯崎 洋 筑波大学, 大学院数理物質科学研究科, 教授 (90111913)
笠原 勇二 筑波大学, 大学院数理物質科学研究科, 教授 (60108975)
三河 寛 筑波大学, 大学院・数理物質科学研究科, 講師 (10219602)
籠屋 恵嗣 筑波大学, 大学院数理物質科学研究科, 助手 (40323258)
|Project Period (FY)
2005 – 2007
Completed (Fiscal Year 2007)
|Budget Amount *help
¥3,500,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥300,000)
Fiscal Year 2007: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2006: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2005: ¥1,300,000 (Direct Cost: ¥1,300,000)
|Keywords||level statistics / random operators / quantum chaos / Anderson model / branching process / Galton-Watson tree / 数理物理 / 確率論 / unfolding / Galton-Watson樹形図 / 局所極限定理|
In this project, we made a general approach to the statistical fluctuation of spectra of random operators from the viewpoint of point process theory, and at the same time, studied the level statistics for a concrete random Schrodinger operator In addition to these researches, we made a probabilistic investigation of random trees, a study motivated by questions of estimating traces of random matrices. Main results are the followings:
1. A mathematical formulation was made of the notion of "unfolding' which is a procedure of normalizing the spectrum of a random operator, so that it can be viewed as a typical realization of a stationary point process.
2. An attempt was made to prove that the unfolded spectrum of the discrete random Schrodinger operator looks like a typical realization of the Poisson point process, and it was clarified that the proof boils down to estimating the probability that two disjoint intervals simultaneously contain eigenvalues of the operator. However, because of the difficulty in obtaining this estimate, the whole proof is still incomplete.
3. Concerning a class of random trees obtained as trajectories of discrete time branching processes, a local limit theorem was obtained for the number of vertices having k children for k=0, 1, 2, .
4. A general extension theorem was obtained for the construction of a probability measure on the space of trees whose vertices have "marks" representing for example spatial location, age, types etc. of vertices.