Random Matrices: Foundations¶

Job: Wigner ensembles and the semicircle law, classical Gaussian ensembles GOE/GUE, edge Tracy–Widom fluctuations, and the one-line bridge to free probability / asymptotic freeness.

Core¶

Empirical spectral distribution¶

For Hermitian (or real symmetric) $H_N\in\mathrm{M}_N$, eigenvalues $\lambda_1\le\cdots\le\lambda_N\in\mathbb{R}$, the empirical spectral distribution (ESD) is $$ \mu_{H_N}=\frac1N\sum_{i=1}^N\delta_{\lambda_i}. $$ Macroscopic RMT studies weak limits of $\mu_{H_N}$ (possibly random $\to$ deterministic). Microscopic RMT rescales gaps to mean spacing $1$ and studies correlation functions.

Wigner matrices¶

A Wigner ensemble (Hermitian case): $H=H^*$, upper-triangular entries $\{H_{ij}:i<j\}$ i.i.d. complex with mean $0$, $\mathbb{E}|H_{ij}|^2=1/N$, diagonal real i.i.d. mean $0$, variance $O(1/N)$, independent of off-diagonal; suitable moment assumptions.

Theorem (Wigner semicircle law). $\mu_{H_N}\Rightarrow\sigma_{\mathrm{sc}}$ almost surely (or in probability, depending on moments), where $$ d\sigma_{\mathrm{sc}}(x)=\frac{1}{2\pi}\sqrt{4-x^2}\,\mathbf{1}_{[-2,2]}(x)\,dx. $$

Proof routes. Method of moments (Wick / Catalan numbers count paths); Stieltjes transform fixed-point $g(z)=\int\frac{d\mu(x)}{x-z}$ satisfies $g= -1/(z+g)$ for semicircle; free probability (below).

Real symmetric Wigner: same semicircle; local statistics fall in GOE class under matching moments.

GOE / GUE / GSE¶

Classical Gaussian ensembles with density proportional to $$ \exp\Bigl(-\frac{\beta N}{4}\operatorname{Tr} H^2\Bigr) $$ on the real vector space of real symmetric ($\beta=1$, GOE), complex Hermitian ($\beta=2$, GUE), or quaternion self-dual ($\beta=4$, GSE) matrices (normalizations vary by author).

Joint eigenvalue density: $$ \frac{1}{Z_{N,\beta}}\prod_{i<j}|\lambda_i-\lambda_j|^\beta \prod_i e^{-\beta N\lambda_i^2/4}\,d\lambda_i. $$ The Vandermonde factor $\prod_{i<j}|\lambda_i-\lambda_j|^\beta$ encodes level repulsion of order $\beta$.

Ensemble	Symmetry	$\beta$
GOE	real symmetric	$1$
GUE	complex Hermitian	$2$
GSE	quaternion self-dual	$4$
CUE	Haar unitary eigenvalues on $S^1$	Dyson circle

Integrable structure (GUE). Correlation functions are determinants of the Hermite kernel; orthogonal polynomial machinery (Deift et al.) yields exact asymptotics.

Bulk universality¶

After unfolding to mean density $1$, $k$-point correlation functions of Wigner matrices (broad entry distributions) converge to the sine kernel process of GUE ($\beta=2$) or GOE/GSE Pfaffian analogs: $$ K_{\mathrm{sine}}(x,y)=\frac{\sin\pi(x-y)}{\pi(x-y)}. $$ Universality philosophy: local spectral statistics depend on symmetry class ($\beta$), not on fine entry details (Erdős–Schlein–Yau, Tao–Vu, …).

Edge: Tracy–Widom¶

Let $\lambda_{\max}$ be the largest eigenvalue of GUE (properly scaled Wigner similarly). Then $$ N^{2/3}\bigl(\lambda_{\max}-2\bigr)\ \Rightarrow\ \mathrm{TW}_2, $$ the Tracy–Widom distribution $F_2(s)=\exp\bigl(-\int_s^\infty (x-s)q(x)^2\,dx\bigr)$ with $q$ the Hastings–McLeod Painlevé II solution. GOE/GSE have $F_1$, $F_4$. Soft-edge Airy kernel replaces sine kernel.

Soft edge vs hard edge: semicircle edge is soft (density vanishes as square root); Wishart hard edge at $0$ involves Bessel kernels.

Free probability bridge (one-liner)¶

Independent Wigner matrices $H_N^{(1)},\ldots,H_N^{(k)}$ (and Haar unitaries) become asymptotically free random variables in the noncommutative probability space $(\mathrm{M}_N,\frac1N\operatorname{Tr})$ as $N\to\infty$ (Voiculescu). Consequently, the limiting ESD of polynomials $p(H_N^{(1)},\ldots)$ is computed by free convolution / $R$-transform calculus — see free probability. Semicircle is the free central limit law; Marchenko–Pastur is free Poisson.

Other classical laws (pointers)¶

Ensemble	Limit ESD
Wishart / sample covariance	Marchenko–Pastur
Haar unitary	circular (eigenvalues on $S^1$ uniform)
Ginibre (non-Hermitian)	circular law in $\mathbb{C}$
Band / sparse	often semicircle still, or tree spectral measures

See it¶

Catalan moments. Semicircle moments: $\int x^{2k}\,d\sigma_{\mathrm{sc}}=\frac{1}{k+1}\binom{2k}{k}$ (Catalan), odd moments $0$. Moment method: $\mathbb{E}\frac1N\operatorname{Tr} H^{2k}$ counts closed walks on complete graphs; leading $N\to\infty$ terms are noncrossing partitions — the same combinatorics as free probability’s free cumulants.

Largest eigenvalue of noise. In spiked covariance models (Johnstone), bulk follows Marchenko–Pastur while outliers pop out beyond the edge; the phase transition (BBP) uses Tracy–Widom fluctuations of the edge as the null distribution for PCA significance tests — RMT as statistical tool.

Check¶

Derive the Stieltjes transform equation for the semicircle law.

$g(z)=\int_{-2}^2 \frac{1}{x-z}\frac{\sqrt{4-x^2}}{2\pi}\,dx$ satisfies $g(z)^2+zg(z)+1=0$ for $z\notin[-2,2]$, with $g(z)\sim -1/z$ at infinity; choose the physical branch $g(z)=(-z+\sqrt{z^2-4})/2$.

Why is the $N^{2/3}$ edge scaling natural from the semicircle density?

Near $x=2-\varepsilon$, density $\rho\sim \sqrt{\varepsilon}/\pi$. Mean eigenvalue spacing at distance $\varepsilon$ from edge is $1/(N\rho)\sim \pi/(N\sqrt{\varepsilon})$. Balancing $\varepsilon\sim 1/(N\rho)$ gives $\varepsilon\sim N^{-2/3}$; fluctuations live on that scale.

Pitfalls¶

Variance normalization differs across books ($1/N$ vs $1/\sqrt{N}$ entries) — radius $2$ vs $2\sqrt{N}$.
ESD convergence does not imply edge universality; edge needs separate rigidity / Green function estimates.
Complex non-Hermitian eigenvalues are not real — circular law, not semicircle.
Single matrix GOE vs two independent Wigner matrices free convolution — freeness is asymptotic independence of NC type, not entrywise independence alone of one matrix’s entries.

Refs¶

Anderson–Guionnet–Zeitouni, An Introduction to Random Matrices
Tao, Topics in Random Matrix Theory
Erdős–Yau, A Dynamical Approach to Random Matrix Theory
Tracy–Widom, “Level-spacing distributions and the Airy kernel”

Next¶

RMT index · Universality & Log-Gases · Determinantal & β-Ensembles · Spiked Models & Statistics · Non-Hermitian & Circular · Free probability foundations · Probability foundations · Mathematical physics · Functional analysis · Representation theory · Operator algebras — spectral measures on Hilbert space