Summary
Some of the statistical properties corresponding to Gaussian ensembles of random matrices whose matrix elements are not all functionally independent are investigated. Three types of ensembles are studied. First, an ensemble whose matrix elements are all real is examined. Next, an ensemble whose matrix elements are not all real, but whose off-diagonal elements have real and imaginary parts which are of the same size on the average is studied. Finally, ensembles with of order-N and-N 2 nonzero imaginary parts of arbitrary size are investigated. Each of the above ensembles is compared with the corresponding Gaussian ensemble in which all of the nonzero matrix elements are functionally independent.
Riassunto
Si esaminano alcune proprietà statistiche corrispondenti a insiemi gaussiani di matrici casuali i cui elementi di matrice non sono tutti funzionalmente indipendenti. Si studiano tre tipi di insieme: prima un insieme i cui elementi di matrice sono tutti reali, poi un insieme i cui elementi di matrice non sono tutti reali, ma i cui elementi al di fuori della diagonale hanno parti reali e immaginarie che sono in media della stessa grandezza, infine insiemi con parti immaginarie diverse da zero di ordineN oN 2 di grandezza arbitraria. Ciascuno dei suddetti insiemi è confrontato con il corrispondente insieme gaussiano nel quale tutti gli elementi di matrice diversi da zero sono funzionalmente indipendenti.
Резюме
Исследуются некоторые статистические свойства, соответствующие гауссовым ансамблям случайных матриц, не все матричные элементы которых являются функционально независимыми. Рассматриваются три типа ансамблей. Сначала исследуется ансамбль, все мчтричные элементы которого являются вещественными. Затем рассматривается ансамбль, у которого не все матричные элементы являются вещественными, но недиагоналяные матричные элементы имеют вещественные н мнимые чссти, которые в среднем являются одинаковыми. В заключение исследуются ансамбли с ненулевыми мнимыми частями порядкаN иN 2. Каждый ие сассмотренных ахсамблей шравнивается с соответствующим гауссовым ансамблем, в котором все ненулевые матричные элемменты являются функционально независимыми.
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This work was supported in part by the National Research Council of Canada.
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McDonald, J.F. Some mathematically simple ensembles of random matrices which represent Hamiltonians with a small time-reversal-noninvariant part. Nuov Cim B 57, 95–124 (1980). https://doi.org/10.1007/BF02722404
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DOI: https://doi.org/10.1007/BF02722404