| Co-Investigator(Kenkyū-buntansha) |
DAN Wakako University of Tsukuba, Institute of Mathematics, Assistant, 数学系, 助手 (40251029)
ISHIKAWA Yasushi University of Tsukuba, Institute of Mathematics, Assistant, 数学系, 助手 (70202976)
TAIRA Kazuaki University of Tsukuba, Institute of Mathematics, Professor, 数学系, 教授 (90016163)
KAJITANI Kunihiko University of Tsukuba, Institute of Mathematics, Professor, 数学系, 教授 (00026262)
AKAHIRA Masafumi University of Tsukuba, Institute of Mathematics, Professor, 数学系, 教授 (70017424)
|
|---|
| Research Abstract |
1. Consider real symmetric matrices X^<(n)>, n<greater than or equal>1 , whose entries are independent and identically distributed Gaussian random varibles. It is well known that the empirical distribution of the eigenvalues of X^<(n)>, under a suitable scaling, converges to the Wigner semi-circle law as n * *. Recently, T.Chan, L.Rogers, Z.Shi, and Y.Takahashi gave alternative proofs to this theorem by considering a fictitious time evolution X^<(n)>(t) of X^<(n)>, which is an idea going back to F.Dyson (1962). They investigated the stochastic differential equation (SDE) satisfied by the n-dimensional diffusion process arising from the eigenvalues of X^<(n)>. However, because of singularity of the coefficients of that SDE, they were involved in an unnecessarily complicated analysis in order to confirm the existence of its solution. We, on the other hand, simplified their argument by noting that as far as we are concerned with the derivation of the semi- circle law, it suffices to obser
… More
ve a simpler SDE satisfied by the Stieltjes transform of the empirical distribution. We also showed that this SDE is easily derived by considering the resolvent of X^<(n)>. These results have been obtained through a joint work with Mr. T.Hiratsuka.
2. We gave a mathematical foundation to the phenomenological side of "level statistics ", which is a common issue in the theories of disordered systems, random matrices, and quantum chaos. This is only possible by regarding the whole or a part of the spectrum of a quantum Hamiltonian as a typical sample of a stationary point process, though this does not seem to have been clearly recognized by physicists. We clarified the following points :
(1)Many of mathematical relations between average quantities appearing in level statistics are corollaries of Palm-Khinchin equality, which is well known in point process theory.
(2)Observing level spacing distribution etc. amounts to observing the Palm measure of the point process, when we regard it as the model of the spectrum. Less
|
