Random Matrix Theory and its Applications

随机矩阵理论及其应用

• 批准号: 04640381
• 负责人: WADATI Miki
• 金额: $ 1.47万
• 依托单位: The University of Tokyo
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for General Scientific Research (C)
• 财政年份: 1992
• 资助国家: 日本
• 起止时间: 1992 至 1994
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-04640381/
• 关键词: Random matrix; Level statistics; Logarithmic interaction; Orthogonal polynomial; Universality; Jastrow wave function; Mesoscopic system; Calogero model; 直交多項式; カオス; メゾスコピック系; Calogero-Moser模型; 相関関数; 量子可積分系; エネルギー準位の相関関数; ジャストロウ型波動関数; ダイソンの

1. A statistical mechanics of one-dimensional particle system with logarithmic interactions is studied. Temperature dependence of the free energy is shown to be universal.2. For ensembles related to classical orthogonal polynomials, the universality of fluctuations of level distributions is proved.3. For the circular ensemble, correlation functions are expressed in terms of orthogonal or skew-orthogonal polynomials.4. Ergodicity of ensembles is proved for ensembles related to classical orthogonal polynomials.5. One-dimensional discrete Schrodinger equation is considered as a model for level statistics. There exists no local fluctuation of level statistics.6. Distribution functions in real space corresponding to Jastrow wavefunctions are expressed in terms of orthogonal or skew-orthogonal polynomials.7. Non-universality of the spectrum at the edge is shown and the spectrum is described by a parameter, which indicates a weak universality at the edge.8. Motion of hamiltonian matrix under a perturbation is discussed.9. Random matrix ensembles related to classical orthogonal polynomials, in particular ones corresponding to exponential-type distributions, are shown to be suitable models for conductivity in mesoscopic systems.10. Relations betweem classical Calogero model and matrix model are discussed.

1. 研究具有对数相互作用的一维粒子系统的统计力学。自由能的温度依赖性被证明是普遍存在的。2．对于与经典正交多项式相关的系综，证明了水平分布涨落的普遍性。 3．对于圆系综，相关函数用正交或斜正交多项式表示。 4．对于与经典正交多项式相关的系综，证明了系综的遍历性。 5．一维离散薛定谔方程被认为是水平统计的模型。水位统计不存在局部波动。6．对应于贾斯特罗波函数的实空间分布函数用正交或斜正交多项式表示。7．表现出边缘谱的非普适性，并用参数描述谱，表明边缘普适性较弱。 8．讨论了扰动下哈密顿矩阵的运动。 9．与经典正交多项式相关的随机矩阵系综，特别是与指数型分布相对应的矩阵系综，被证明是介观系统中电导率的合适模型。10．讨论了经典Calogero模型与矩阵模型之间的关系。

项目成果

T.Nagao: "An Integration Method on Generalized Circular Ensembles" J.Phys.Soc.Jpn.61. 1903-1909 (1992)

T.Nagao：“广义圆形系综的积分方法”J.Phys.Soc.Jpn.61。

T.Nagao: "Eigenvalue Distribution of Random Matrices at the Spectrum Edge" J.Phys.Soc.Jpn.62. 3845-3856 (1993)

T.Nagao：“频谱边缘随机矩阵的特征值分布”J.Phys.Soc.Jpn.62。

T.Nagao: "Correlation Functions for Jastrow-Product Wave Functions" Journal of Physical Society of Japan. 62. 480-488 (1993)

T.Nagao：“Jastrow 积波函数的相关函数”日本物理学会杂志。

K.Slevin: "New Random Matrix Theory of Scattering in Mesoscopic Systems" Phys.Rev.Lett.70. 635-638 (1993)

K.Slevin：“介观系统散射的新随机矩阵理论”Phys.Rev.Lett.70。

M.Shiroishi: "Level Spacing Distributions of Random Matrix Ensembles" J.Phys.Soc.Jpn.62. 2248-2259 (1993)

M.Shiroishi：“随机矩阵系综的水平间距分布”J.Phys.Soc.Jpn.62。