TheoremProved

Wigner's semicircle law

Statement

Let $W_n = (X_{ij})_{1 \le i,j \le n}$ be a sequence of $n \times n$ real symmetric (or complex Hermitian) random matrices whose upper-triangular entries $X_{ij}$ ( $i \le j$ ) are independent random variables with mean $0$ , off-diagonal variance $1$ , and bounded higher moments. Let $\lambda_1 \le \dots \le \lambda_n$ be the eigenvalues of $\frac{1}{\sqrt{n}}W_n$ . As $n \to \infty$ , the empirical spectral measure $\mu_n = \frac{1}{n}\sum_{k=1}^n \delta_{\lambda_k}$ converges weakly almost surely to the Wigner semicircle distribution with density $\rho_{\mathrm{sc}}(x) = \frac{1}{2\pi}\sqrt{4 - x^2}\,\mathbf{1}_{[-2,2]}(x)$ .

Why is it true?

Just as the central limit theorem says that the sum of many independent random numbers follows a universal bell curve regardless of their individual distributions, Wigner's semicircle law says that the eigenvalues of a large symmetric matrix filled with independent random entries spread out into a universal semicircular arch on $[-2,2]$ once rescaled by $\sqrt{n}$ .

Proof sketch

By the method of moments, one computes the expected trace moments $\int x^k\,d\mu_n(x) = \frac{1}{n}\operatorname{Tr}\left(\left(\frac{W_n}{\sqrt{n}}\right)^k\right) = n^{-1 - k/2} \sum_{i_1, \dots, i_k = 1}^n X_{i_1 i_2} X_{i_2 i_3} \cdots X_{i_k i_1}$ . Each term corresponds to a closed walk of length $k$ on $\{1, \dots, n\}$ . Because $\mathbb{E}[X_{ij}] = 0$ , any walk traversing an edge only once has expectation $0$ . For odd $k$ , all dominant contributions vanish in the limit $n \to \infty$ . For even $k = 2m$ , the only walks surviving the $n^{-1-m}$ normalization are double trees on $m + 1$ vertices that traverse each of $m$ edges exactly twice; the number of such canonical walks is the Catalan number $C_m = \frac{1}{m+1}\binom{2m}{m}$ . Since $\int_{-2}^2 x^{2m} \rho_{\mathrm{sc}}(x)\,dx = C_m$ and $\rho_{\mathrm{sc}}$ is compactly supported, convergence of moments uniquely determines weak convergence to $\rho_{\mathrm{sc}}$ .

Topics that use this theorem

Random matrix theory

Related theorems

Central limit theorem Spectral theorem (real symmetric matrices)

Step-by-step proofs

No step-by-step proof yet for this theorem.

References

Eugene P. Wigner (1955). Characteristic Vectors of Bordered Matrices with Infinite Dimensions
Eugene P. Wigner (1958). On the Distribution of the Roots of Certain Symmetric Matrices
Greg W. Anderson, Alice Guionnet, Ofer Zeitouni (2010). An Introduction to Random Matrices