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I will start with a little bit of history by posing the problem that confronted Wigner: that of predicting the distribution of spectral lines of the nucleus. After a building some intuition for what this distribution looks like, I'll go through a rigorous proof of the Wigner surmise (the distribution of eigenvalues of a random matrix) for a 2x2 matrix. Through numerical simulations, I will try to convince you that this result works quite well for larger NxN matrices as well. Then, I will compare it to a distribution of spacings between randomly generated points and try to point out some unique features of the distribution of eigenvalues of a random matrix.
Board Notes:
The notes contains derivations of functional forms of the distributions for:
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