A theorem due to Stielties shows that the problem of locating the zeros of the classical polynomials is equivalent to finding the electrostatic equilibrium positions for a set of interacting point charges. Defining a density function for these same charges we find that the density of zeros for the *n*th Hermite polynomial is the same as the density of eigenvalues for an ensemble of *n*‐dimensional Hermitian matrices. Similarly the location of the zeros of the *n*th Laguerre polynomial determines the density of eigenvalues for an ensemble of *n*‐dimensional positive matrices, and the zeros of the ½*n*th Tchebichef polynomial determine the density for the real part of the eigenvalues for an ensemble of *n*‐dimensional unitary matrices.

## REFERENCES

1.

(a) E. P. Wigner, “Distribution Laws for Roots of a Random Hermitian Matrix” (unpublished). The pertinent results and definitions are available in

(b) N. Rosenzweig, “Brandeis Summer Institute 1962, Statistical Physics” (W. A. Benjamin, Inc., New York, 1963).

2.

B. V. Bronk, “Exponential Ensemble for Random Matrices,” J. Math. Phys. (to be published).

3.

F. J.

Dyson

, J. Math. Phys.

3

, 140

(1962

). Notice the results of the present paper apply to all cases of Dyson’s classification of ensembles by transformation properties. The formulas here are stated for the case $\beta \u2009=\u20092.$4.

*Bateman Manuscript Project*, edited by A. Erdléyi (McGraw‐Hill Book Company, Inc., New York, 1953). See (a) Sec. 10.13; (b) Sec. 10.12; (c) Sec. 10.11.

5.

6.

From this point on the proof also applies to Rosenzweig’s “Fixed Strength” ensemble, see Ref. 1(b), p. 110.

7.

E. P. Wigner,

*Proceedings of the Canadian Mathematical Congress*, (1954), p. 174 (unpublished).8.

M. Born,

*Mechanics of the Atom*, translated by Fisher and Hartree, (G. Bell and Sons, London, 1960), Appendix II.9.

10.

We have actually proved for ensemble III a special case of a much more general theorem, which can be stated roughly, that the zeros of a set of polynomials orthogonalized with respect to an

*arbitrary*weight function on $[\u22121,1],$ are cosines of angles distributed uniformly around the circle. See Ref. 5, Theorem 12.7.2.
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© 1964 The American Institute of Physics.

1964

The American Institute of Physics

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