Summary
The fundamental solution for the Brownian motion of the eigenvalues of a random matrix in Dyson’s (1) model is represented as an integral with respect to the measure on the orthogonal group inN dimensions. The solution is discussed for some special cases only. In two dimensions the distribution of the spacing between the eigenvalues is calculated as a function of the time and is found to depend only upon the initial value of thespacing.
Riassunto
Si rappresenta la soluzione fondamentale per il moto browniano degli autovalori di una matrice casuale nel modello di Dyson (1) in forma di un integrale rispetto alla misura su un gruppo ortogonale inN dimensioni. Si discute la soluzione solo per alcuni casi speciali. In due dimensioni, la distribuzione degli intervalli fra gli autovalori si calcola in funzione del tempo e risulta dipendere soltanto dal valore iniziale dellaspaziatura.
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Eesearch supported by the Weizmann Memorial Foundation and by the U. S. Atomic Energy Commission.
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Rosenzweig, F. On the Brownian-motion model for the eigenvalues of a random matrix. Nuovo Cim 38, 1047–1053 (1965). https://doi.org/10.1007/BF02748615
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DOI: https://doi.org/10.1007/BF02748615