"Gaussian Orthogonal Ensemble"의 두 판 사이의 차이
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imported>Pythagoras0 |
imported>Pythagoras0 |
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| 4번째 줄: | 4번째 줄: | ||
on the space of $n\times n$ real symmetric matrices $H=(H_{ij})$ | on the space of $n\times n$ real symmetric matrices $H=(H_{ij})$ | ||
* Its distribution is invariant under orthogonal conjugation, and it models Hamiltonians with time-reversal symmetry | * Its distribution is invariant under orthogonal conjugation, and it models Hamiltonians with time-reversal symmetry | ||
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| + | ==level spacing of eigenvalues== | ||
| + | * From the ordered sequence of eigenvalues <math>\lambda_1 < \ldots < \lambda_n < \lambda_{n+1} < \ldots</math>, one defines the normalized spacings <math>s = (\lambda_{n+1} - \lambda_n)/\langle s \rangle</math>, where <math>\langle s \rangle =\langle \lambda_{n+1} - \lambda_n \rangle</math> is the mean spacing. | ||
| + | * The probability distribution of spacings is approximately given by, | ||
| + | : <math> p_1(s) = \frac{\pi}{2}s\, \mathrm{e}^{-\frac{\pi}{4} s^2} </math> | ||
| + | for the orthogonal ensemble GOE <math>\beta=1</math> | ||
==computational resource== | ==computational resource== | ||
* https://drive.google.com/file/d/0B8XXo8Tve1cxZEdFLUVsM0hwUWM/view | * https://drive.google.com/file/d/0B8XXo8Tve1cxZEdFLUVsM0hwUWM/view | ||
2016년 6월 29일 (수) 00:50 판
introduction
- The Gaussian orthogonal ensemble GOE(n) is described by the Gaussian measure with density
\[ \frac{1}{Z_{\text{GOE}(n)}} e^{- \frac{n}{4} \mathrm{tr} H^2} \] on the space of $n\times n$ real symmetric matrices $H=(H_{ij})$
- Its distribution is invariant under orthogonal conjugation, and it models Hamiltonians with time-reversal symmetry
level spacing of eigenvalues
- From the ordered sequence of eigenvalues \(\lambda_1 < \ldots < \lambda_n < \lambda_{n+1} < \ldots\), one defines the normalized spacings \(s = (\lambda_{n+1} - \lambda_n)/\langle s \rangle\), where \(\langle s \rangle =\langle \lambda_{n+1} - \lambda_n \rangle\) is the mean spacing.
- The probability distribution of spacings is approximately given by,
\[ p_1(s) = \frac{\pi}{2}s\, \mathrm{e}^{-\frac{\pi}{4} s^2} \] for the orthogonal ensemble GOE \(\beta=1\)