"Gaussian Orthogonal Ensemble"의 두 판 사이의 차이
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| 2번째 줄: | 2번째 줄: | ||
* The Gaussian orthogonal ensemble GOE(n) is described by the Gaussian measure with density | * The Gaussian orthogonal ensemble GOE(n) is described by the Gaussian measure with density | ||
:<math> \frac{1}{Z_{\text{GOE}(n)}} e^{- \frac{n}{4} \mathrm{tr} H^2} </math> | :<math> \frac{1}{Z_{\text{GOE}(n)}} e^{- \frac{n}{4} \mathrm{tr} H^2} </math> | ||
| − | on the space of | + | on the space of <math>n\times n</math> real symmetric matrices <math>H=(H_{ij})</math> |
* 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 | ||
| 16번째 줄: | 16번째 줄: | ||
* The joint probability density for the eigenvalues ''λ''<sub>1</sub>,''λ''<sub>2</sub>,...,''λ''<sub>''n''</sub> of GOE is given by | * The joint probability density for the eigenvalues ''λ''<sub>1</sub>,''λ''<sub>2</sub>,...,''λ''<sub>''n''</sub> of GOE is given by | ||
: <math>\frac{1}{Z_{\beta, n}} \prod_{k=1}^n e^{-\frac{\beta n}{4}\lambda_k^2}\prod_{i<j}\left|\lambda_j-\lambda_i\right|^\beta~, \quad (1)</math> | : <math>\frac{1}{Z_{\beta, n}} \prod_{k=1}^n e^{-\frac{\beta n}{4}\lambda_k^2}\prod_{i<j}\left|\lambda_j-\lambda_i\right|^\beta~, \quad (1)</math> | ||
| − | where the Dyson index | + | where the Dyson index <math>\beta=1</math>, counts the number of real components per matrix element; ''Z''<sub>''β'',''n''</sub> is a normalisation constant which can be explicitly computed as Selberg integral. |
* eigenvalues repel as the joint probability density has a zero (of <math>\beta</math>th order) for coinciding eigenvalues <math>\lambda_j=\lambda_i</math>. | * eigenvalues repel as the joint probability density has a zero (of <math>\beta</math>th order) for coinciding eigenvalues <math>\lambda_j=\lambda_i</math>. | ||
2020년 11월 13일 (금) 19:24 기준 최신판
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 \(\beta=1\)
joint eigenvalue distribution
- The joint probability density for the eigenvalues λ1,λ2,...,λn of GOE is given by
\[\frac{1}{Z_{\beta, n}} \prod_{k=1}^n e^{-\frac{\beta n}{4}\lambda_k^2}\prod_{i<j}\left|\lambda_j-\lambda_i\right|^\beta~, \quad (1)\] where the Dyson index \(\beta=1\), counts the number of real components per matrix element; Zβ,n is a normalisation constant which can be explicitly computed as Selberg integral.
- eigenvalues repel as the joint probability density has a zero (of \(\beta\)th order) for coinciding eigenvalues \(\lambda_j=\lambda_i\).