Wigner Semicircle Law

4 Universal Distribution

The Law: For a large symmetric random matrix with independent entries (mean 0, variance 1), the normalized eigenvalues $\lambda/\sqrt{n}$ distribute according to: $$ \rho(x) = \frac{1}{2\pi} \sqrt{4 - x^2}, \quad |x| \le 2 $$
Experiment: The following demonstration illustrates how the histogram of the eigenvalues of a random matrix behave as the matrix size increases.
Demo Details: Randomly sampling eigenvalues of a $n\times n$ symmetric matrix whose entries are $\pm 1$ with probability $1/2$. Then plotting the histogram of the eigenvalues after scaling $1/\sqrt{n}$

EXPERIMENT Convergence Simulator

Randomly sampling eigenvalues of a $n\times n$ symmetric matrix whose entries are $\pm 1$ with probability $1/2$. Then plotting the histogram of the eigenvalues after scaling $1/\sqrt{n}$

Samples ($n$) 100

100 3000 (Max)

Sample Stats

Note: The red curve represents the theoretical limit density. The blue bars are the empirical histogram of our current sample.