Abstract
Dyson's two-dimensional Coulomb gas on the unit circle with inverse temperature beta is investigated with the help of a Metropolis algorithm. Theoretical predictions for energy and specific heat are verified. The connection with the theory of ensembles of random unitary matrices with orthogonal, unitary or symplectic symmetry corresponding to beta =1, 2, 4 is investigated in detail. New approximation formulae for the spacing distribution and the so-called delta statistics are proposed and found to be useful, not only in the three cases mentioned but also for all beta between 0 and 4.
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