Girko's Circular Law

Let  be (possibly complex) eigenvalues of a set of random  real matrices with entries independent and taken from a standard normal distribution. Then as ,  is uniformly distributed on the unit disk in the complex plane. For small , the distribution shows a concentration along the real line accompanied by a slight paucity above and below (with interesting embedded structure). However, as , the concentration about the line disappears and the distribution becomes truly uniform.

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Girko, V. L. Theory of Random Determinants. Boston, MA: Kluwer, 1990.

Mehta, M. L. Random Matrices, 3rd ed. New York: Academic Press, 2004.