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In the theory of random matrices, the circular ensembles are measures on spaces of unitary matrices introduced by Freeman Dyson as modifications of the Gaussian matrix ensembles.
Together with Leonid Pastur, Vladimir Marchenko discovered the Marchenko–Pastur law in random matrix theory.
In subsequent work, Siegel made contributions to physicist's understanding of duality, conformal invariance, AdS/CFT, the random matrix approach to string theory, twistor superstrings, and other topics in string theory.
He has contributed to one of the discoveries concerning the Riemann zeta function, namely, that the Riemann zeros appear to display the same statistics as those which are believed to be present in energy levels of quantum chaotic systems and described by Random Matrix Theory.