Math 2.0

Probability theory and stochastic processes Theorem

Wigner semicircle law

Builds on Hermitian matrixEigenvalueIndependent and identically distributed random variablesConvergence in distribution

Statement

Let (Yij)1≤i<j(Y_{ij})_{1 \le i < j} be independent and identically distributed real random variables with mean 00 and variance 11. Let (Yii)i≥1(Y_{ii})_{i \ge 1} be independent and identically distributed real random variables with mean 00 and finite variance, independent of the YijY_{ij} with i<ji < j. For n≥1n \ge 1, let WnW_n be the real symmetric n×nn \times n matrix with entries Wn(i,j)=Wn(j,i)=Yij/nW_n(i, j) = W_n(j, i) = Y_{ij} / \sqrt n for i≤ji \le j. Let λ1,…,λn\lambda_1, \ldots, \lambda_n be the eigenvalues of WnW_n, and let μn=1n∑k=1nδλk\mu_n = \frac1n \sum_{k=1}^n \delta_{\lambda_k} be its empirical spectral measure. Then, almost surely, μn\mu_n converges weakly as n→∞n \to \infty to the semicircle distribution, the probability measure on ℝ\R with density ρ(x)=12π4−x2for ∣x∣≤2,ρ(x)=0for ∣x∣>2.\rho(x) = \frac{1}{2\pi} \sqrt{4 - x^2} \quad \text{for } \abs{x} \le 2, \qquad \rho(x) = 0 \quad \text{for } \abs{x} > 2.

Remarks

The moment method is the classical proof. One shows that 𝔼∫xk dμn=1n𝔼tr⁡Wnk\mathbb{E} \int x^k \, d\mu_n = \frac1n \mathbb{E} \tr W_n^k tends to the kk-th moment of the semicircle law, which is 00 for odd kk and the Catalan number 1m+1(2mm)\frac{1}{m+1}\binom{2m}{m} for k=2mk = 2m. Expand the trace as a sum over closed paths. In the limit only paths that trace a tree and cross each edge exactly twice contribute, and Catalan numbers count them. A variance bound and a truncation of the entries then give almost sure convergence. The same result holds for complex Hermitian matrices with independent and identically distributed entries above the diagonal, of mean 00 and 𝔼∣Yij∣2=1\mathbb{E}\abs{Y_{ij}}^2 = 1.

Going deeper

For a student

It is the law of large numbers of random matrix theory, the analogue for spectra of the law of large numbers for averages. Wigner introduced such matrices as models for the energy levels of heavy atomic nuclei. Finer questions, such as the fluctuations of the largest eigenvalue, lead to the Tracy–Widom distribution. For sample covariance matrices the analogue is the Marchenko–Pastur law.

Reading path

To understand this page from the ground up, read these 9 pages in order. The foundations come first.

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Convergence in distribution, concept, no evidence levelConvergence in distri…conceptEigenvalue, concept, no evidence levelEigenvalueconceptIndependent and identically distributed random variables, concept, no evidence levelIndependent and ident…conceptMatrix, definition, no evidence levelMatrixdefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionTranspose, definition, no evidence levelTransposedefinitionConjugate transpose, definition, no evidence levelConjugate transposedefinitionHermitian matrix, definition, no evidence levelHermitian matrixdefinitionWigner semicircle law, statement, no evidence levelWigner semicircle lawstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Convergence in distribution concept
  2. Eigenvalue concept
  3. Independent and identically distributed random variables concept
  4. Matrix definition
  5. Square matrix definition
  6. Transpose definition
  7. Conjugate transpose definition
  8. Hermitian matrix definition
  9. Wigner semicircle law statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Wigner_semicircle_distribution
About this page
Tag3QTT95DMFH
URIm2://3QTT95DMFH
Kindstatement, theorem
WritingW0 stub
MSC60B20, 15B52, 60F05
StatusThis copy of the site has no build data. Run m2 status 3QTT95DMFH.
AboutEigenvalue
UsesHermitian matrix, Eigenvalue, Independent and identically distributed random variables, Convergence in distribution
Changed2 November 2026

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