Gaussian ensemble

In random matrix theory, the Gaussian ensembles are the most-commonly studied ensembles. They are often denoted by their Dyson index, β = 1 for GOE, β = 2 for GUE, and β = 4 for GSE. This index counts the number of real components per matrix element.

It is also called the Wigner ensemble, or the Hermite ensemble.

Definitions

Conventions

There are many conventions for defining the Gaussian ensembles. In this article, we specify exactly one of them.

In all definitions, the Gaussian ensemble have zero expectation.

• $\beta$: a positive real number. Called the Dyson index. The cases of $\beta =1,2,4$ are special.
• $N$: the side-length of a matrix. Always a positive integer.
• $W_{N}$: a matrix sampled from a Gaussian ensemble with size $N\times N$. The letter $W$ stands for "Wigner".
• $M^{*}$: the adjoint of a matrix. We assume $W_{N}=W_{N}^{*}$ (self-adjoint) when $W_{N}$ is sampled from a gaussian ensemble.
  • If $M$ is real, then $M^{*}$ is its transpose.
  • If $M$ is complex or quaternionic, then $M^{*}$ is its conjugate transpose.
• $\lambda _{1},\dots ,\lambda _{N}$: the eigenvalues of the matrix, which are all real, since the matrices are always assumed to be self-adjoint.
• $\sigma _{d}^{2}$: the variance of on-diagonal matrix entries. We assume that for each $N$, all on-diagonal matrix entries have the same variance. It is always defined as $\mathbb {E} [|W_{N,}|^{2}]$.
• $\sigma _{od}^{2}$: the variance of off-diagonal matrix entries. We assume that for each $N$, all off-diagonal matrix entries have the same variance. It is always defined as $\mathbb {E} [|W_{N,ij}|^{2}]$ where $i\neq j$.
  • For a complex number, $|a+bi|^{2}=a^{2}+b^{2}$.
  • For a quaternion, $|a+bi+cj+dk|^{2}=a^{2}+b^{2}+c^{2}+d^{2}$.
• $Z$: the partition function.

Summary of convention in the page

Name	GOE(N)	GUE(N)	GSE(N)	GβE(N)
Full name	Gaussian orthogonal ensemble	Gaussian unitary ensemble	Gaussian symplectic ensemble	Gaussian beta ensemble
$\beta$	1	2	4	β
$\sigma _{d}^{2}$	2	1	1/2	2/β
$\sigma _{od}^{2}$	1	1	1	1
matrix density	${\frac {1}{Z}}e^{-{\frac {1}{4}}\mathrm {Tr} W_{N}^{2}}$	${\frac {1}{Z}}e^{-{\frac {1}{2}}\mathrm {Tr} W_{N}^{2}}$	${\frac {1}{Z}}e^{-\mathrm {Tr} W_{N}^{2}}$	${\frac {1}{Z}}e^{-{\frac {1}{4}}\beta \mathrm {Tr} W_{N}^{2}}$
$Z$	$2^{{\frac {1}{4}}N(N+3)}\pi ^{{\frac {1}{4}}N(N+1)}$	$2^{{\frac {1}{2}}N}\pi ^{{\frac {1}{2}}N^{2}}$	$2^{-N(N-1)}\pi ^{{\frac {1}{2}}N(2N-1)}$	$2^{{\frac {1}{2}}N}\left({\frac {2\pi }{\beta }}\right)^{{\frac {1}{2}}N+{\frac {1}{4}}\beta N(N-1)}$

When referring to the main reference works, it is necessary to translate the formulas from them, since each convention leads to different constant scaling factors for the formulas.

Conventions in reference works

Name	$\sigma _{d}^{2}$	$\sigma _{od}^{2}$
Wikipedia (this page)	2/β	1
(Deift 2000) (β = 2 only)	1/2	1/2
(Mehta 2004)	1/β	1/2
(Anderson, Guionnet & Zeitouni 2010)	2/β	1
(Forrester 2010) for β = 1, 2, 4	1/β	1/2
(Forrester 2010) for β ≠ 1, 2, 4	1	β/2
(Tao 2012) (β = 2 only)	1	1
(Mingo & Speicher 2017) (β = 2 only)	1/N	1/N
(Livan, Novaes & Vivo 2018)	1	β/2
(Potters & Bouchaud 2020)	${\frac {2\sigma ^{2}}{\beta N}}$	${\frac {\sigma ^{2}}{N}}$

There are equivalent definitions for the GβE(N) ensembles, given below.

By sampling

For all $\beta =1,2,4$ cases, the GβE(N) ensemble is defined by how it is sampled:

• Sample a gaussian matrix $X_{N}$, such that all its entries are IID sampled from the corresponding standard normal distribution.
  • If $\beta =1$, then $X_{N,ij}\sim {\mathcal {N}}(0,1)$.
  • If $\beta =2$, then $X_{N,ij}\sim {\mathcal {N}}(0,1/2)+i{\mathcal {N}}(0,1/2)$.
  • If $\beta =4$, then $X_{N,ij}\sim {\mathcal {N}}(0,1/4)+i{\mathcal {N}}(0,1/4)+j{\mathcal {N}}(0,1/4)+k{\mathcal {N}}(0,1/4)$.
• Let $W_{N}={\frac {1}{\sqrt {2}}}(X+X^{*})$.

By density

For all $\beta =1,2,4$ cases, the GβE(N) ensemble is defined with density function $$\rho (W_{N})={\frac {1}{Z}}e^{-{\frac {\beta }{4}}\sum _{i=1}^{N}W_{N,ii}^{2}-{\frac {\beta }{2}}\sum _{1\leq i<j\leq N}|W_{N,ij}|^{2}}={\frac {1}{Z}}e^{-{\frac {\beta }{4}}\mathrm {Tr} W_{N}^{2}}$$where the partition function is $Z=2^{{\frac {1}{2}}N}\left({\frac {2\pi }{\beta }}\right)^{{\frac {1}{2}}N+{\frac {1}{4}}\beta N(N-1)}$.

The Gaussian orthogonal ensemble GOE(N) is defined as the probability distribution over $N\times N$ symmetric matrices with density function$$\rho (W_{N})={\frac {1}{Z}}e^{-{\frac {1}{4}}\sum _{i=1}^{N}W_{N,ii}^{2}-{\frac {1}{2}}\sum _{1\leq i<j\leq N}W_{N,ij}^{2}}={\frac {1}{Z}}e^{-{\frac {1}{4}}\mathrm {Tr} W_{N}^{2}}$$where the partition function is $Z=2^{{\frac {1}{4}}N(N+3)}\pi ^{{\frac {1}{4}}N(N+1)}$.

Explicitly, since there are only ${\frac {1}{2}}N(N+1)$ degrees of freedom, the parameterization is as follows:$$\rho (W_{N})\prod _{1\leq i\leq j\leq N}dW_{N,ij}$$where we pick the upper diagonal entries $\{W_{ij}\}_{1\leq i\leq j\leq N}$ as the degrees of freedom.

The Gaussian unitary ensemble GUE(N) is defined as the probability distribution over $N\times N$ Hermitian matrices with density function$$\rho (W_{N})={\frac {1}{Z}}e^{-{\frac {1}{2}}\sum _{i=1}^{N}W_{N,ii}^{2}-\sum _{1\leq i<j\leq N}|W_{N,ij}|^{2}}={\frac {1}{Z}}e^{-{\frac {1}{2}}\mathrm {Tr} \,W_{N}^{2}}.$$where the partition function is $Z=2^{{\frac {1}{2}}N}\pi ^{{\frac {1}{2}}N^{2}}$.

Explicitly, since there are only $N^{2}$ degrees of freedom, the parameterization is as follows: $$\rho (W_{N})\,\prod _{i=1}^{N}dW_{N,ii}\;\prod _{1\leq i<j\leq N}d(\mathrm {Re} \,W_{N,ij})\,d(\mathrm {Im} \,W_{N,ij})$$where we pick the upper diagonal entries $\{W_{N,ii}\}_{1\leq i\leq N}\cup \{\mathrm {Re} \,W_{N,ij},\,\mathrm {Im} \,W_{N,ij}\}_{1\leq i<j\leq N}$ as the degrees of freedom.

The Gaussian symplectic ensemble GSE(N) is defined as the probability distribution over $N\times N$ self‑adjoint quaternionic matrices with density function$$\rho (W_{N})={\frac {1}{Z}}e^{-\sum _{i=1}^{N}W_{N,ii}^{2}-2\sum _{1\leq i<j\leq N}|W_{N,ij}|^{2}}={\frac {1}{Z}}e^{-\mathrm {Tr} \,W_{N}^{2}}.$$where the partition function is $Z=2^{-N(N-1)}\pi ^{{\frac {1}{2}}N(2N-1)}$.

Explicitly, since there are only $N(2N-1)$ degrees of freedom, the parameterization is as follows:$$\rho (W_{N})\,\prod _{i=1}^{N}dW_{N,ii}\;\prod _{1\leq i<j\leq N}\prod _{a=0}^{3}dW_{N,ij}^{(a)}$$where we write $W_{N,ij}=W_{N,ij}^{(0)}+i\,W_{N,ij}^{(1)}+j\,W_{N,ij}^{(2)}+k\,W_{N,ij}^{(3)}$ and pick the upper diagonal entries $\{W_{N,ii}\}_{1\leq i\leq N}\cup \{W_{N,ij}^{(a)}\}_{1\leq i<j\leq N,\;0\leq a\leq 3}$ as the degrees of freedom.

By invariance

For all $\beta =1,2,4$ cases, the GβE(N) ensemble is uniquely characterized (up to affine transform) by its symmetries, or invariance under appropriate transformations.

For GOE, consider a probability distribution over $N\times N$ symmetric matrices satisfying the following properties:

• Invariance under orthogonal transformation: For any fixed (not random) $N\times N$ orthogonal matrix $O$, let $M$ be a random sample from the distribution. Then $OMO^{T}$ has the same distribution as $M$.
• Independence: The entries $\{M_{ij}\}_{1\leq i\leq j\leq N}$ are independently distributed.

For GUE, consider a probability distribution over $N\times N$ Hermitian matrices satisfying the following properties:

• Invariance under unitary transformation: For any fixed (not random) $N\times N$ unitary matrix $U$, let $M$ be a random sample from the distribution. Then $UMU^{*}$ has the same distribution as $M$.
• Independence: The entries $\{M_{ij}\}_{1\leq i\leq j\leq N}$ are independently distributed.

For GSE, consider a probability distribution over $N\times N$ self-adjoint quaternionic matrices satisfying the following properties:

• Invariance under symplectic transformation: For any fixed (not random) $N\times N$ symplectic matrix $S$, let $M$ be a random sample from the distribution. Then $SMS^{*}$ has the same distribution as $M$.
• Independence: The entries $\{M_{ij}\}_{1\leq i\leq j\leq N}$ are independently distributed.

In all 3 cases, these conditions force the distribution to have the form $\rho (M)={\frac {1}{Z}}e^{-a\operatorname {Tr} (M^{2})+b\operatorname {Tr} (M)}$, where $a>0$ and $b,Z\in \mathbb {R}$. Thus, with the further specification of ${\frac {1}{N}}\mathbb {E} [\operatorname {Tr} (M)]=0,{\frac {1}{N}}\mathbb {E} [\operatorname {Tr} (M^{2})]={\frac {2}{\beta }}$, we recover the GOE, GUE, GSE.

More succinctly stated, each of GOE, GUE, GSE is uniquely specified by invariance, independence, the mean, and the variance.

By spectral distribution

For all $\beta =1,2,4$ cases, the GβE(N) ensemble is defined as the ensemble obtained by $ADA^{*}$, where

• $D=\operatorname {diag} (\lambda _{1},\dots ,\lambda _{N})$ is a diagonal real matrix with its entries sampled according to the spectral density, defined below;
• $A$ is an orthogonal/unitary/symplectic matrix sampled uniformly, that is, from the normalized Haar measure of the orthogonal/unitary/symplectic group.

Spectral density

For eigenvalues $\lambda _{1},\dots ,\lambda _{N}$ the joint density of GβE(N) is$$\rho _{\beta ,N}(\lambda _{1},\dots ,\lambda _{N})={\frac {1}{Z_{\beta ,N}}}e^{-{\frac {\beta }{4}}\sum _{i=1}^{N}\lambda _{i}^{2}}\prod _{1\leq i<j\leq N}|\lambda _{i}-\lambda _{j}|^{\beta }={\frac {1}{Z_{\beta ,N}}}e^{-{\frac {\beta }{4}}\|\lambda \|_{2}^{2}}|\Delta _{N}(\lambda )|^{\beta }$$where $\Delta _{N}$ is the Vandermonde determinant, and the partition function $Z_{\beta ,N}$ is explicitly evaluated as a Selberg integral:$${\begin{aligned}Z_{\beta ,N}&=\int _{\mathbb {R} ^{N}}e^{-{\frac {\beta }{4}}\sum _{i=1}^{N}\lambda _{i}^{2}}\prod _{1\leq i<j\leq N}|\lambda _{i}-\lambda _{j}|^{\beta }d\lambda \\&=(2\pi )^{\frac {N}{2}}\left({\frac {2}{\beta }}\right)^{{\frac {1}{2}}N+{\frac {1}{4}}\beta N(N-1)}\prod _{j=1}^{N}{\frac {\Gamma \left(1+j{\frac {\beta }{2}}\right)}{\Gamma \left(1+{\frac {\beta }{2}}\right)}}\end{aligned}}$$where $\Gamma$ is the Euler Gamma function.

Determinantal point process

Histogram of the empirical spectral density of ${\frac {1}{\sqrt {N}}}W_{N}$ for $2^{0},\dots ,2^{5}$, obtained by averaging over $10^{5}$ samples of the matrix, compared with the theoretical prediction of ${\sqrt {N}}K_{N}({\sqrt {N}}x,{\sqrt {N}}x)$.

Define functions $\psi _{n}(x):={\frac {e^{-{\frac {1}{4}}x^{2}}}{\sqrt {n!{\sqrt {2\pi }}}}}\operatorname {He} _{n}(x)$, where $\operatorname {He}$ is the probabilist's Hermite polynomial. These are the wavefunction states of the quantum harmonic oscillator.

The spectrum of GUE(N) is a determinantal point process with kernel $K_{N}(x,x'):=\sum _{n=0}^{N-1}\psi _{n}(x)\psi _{n}(x')$.

This allows a closed-form solution of the spectral density of GUE(N) for finite values of $N$:$$\rho (x)={\frac {1}{N}}K_{N}(x,x)={\frac {1}{N{\sqrt {2\pi }}}}e^{-{\frac {1}{2}}x^{2}}\sum _{j=0}^{N-1}{\frac {1}{j!}}\operatorname {He} _{j}(x)^{2}$$The spectral distribution of $\beta =1,4$ can also be written as a quaternionic determinantal point process involving skew-orthogonal polynomials.

Tridiagonalization

For all $\beta =1,2,4$ cases, given a sampled matrix $W_{N}$ from the GβE(N) ensemble, we can perform a Householder transformation tridiagonalization on it to obtain a tridiagonal matrix $T_{\beta ,N}$, which has the same distribution as$${\sqrt {\frac {2}{\beta }}}{\begin{bmatrix}a_{N}&b_{N-1}/{\sqrt {2}}&0&\cdots &0\\b_{N-1}/{\sqrt {2}}&a_{N-1}2&b_{N-2}/{\sqrt {2}}&\ddots &\vdots \\0&b_{N-2}/{\sqrt {2}}&\ddots &\ddots &0\\\vdots &\ddots &\ddots &a_{2}&b_{1}/{\sqrt {2}}\\0&\cdots &0&b_{1}/{\sqrt {2}}&a_{1}\end{bmatrix}}$$where each $a_{1},\dots ,a_{N}\sim {\mathcal {N}}(0,1)$ is gaussian-distributed, and each $b_{i}\sim \chi _{i\beta }$ is chi-distributed, and all $a_{1},\dots ,a_{N},b_{1},\dots ,b_{N-1}$ are independent.

In particular, this definition allows extension to all $\beta >0$ cases, leading to the gaussian beta ensembles.

Global law

Spectral density of GOE/GUE/GSE for $N=2^{0},2^{1},\dots ,2^{5}$. Each distribution is normalized to converge to the semicircle with radius 2.

The Wigner semicircle law states that the empirical eigenvalue distribution of ${\frac {1}{\sqrt {N}}}W_{N}$ converges in distribution to the Wigner semicircle distribution with radius 2. That is, the distribution on $[-2,+2]$ with probability density function $$\rho _{sc}(x)={\frac {\sqrt {4-x^{2}}}{2\pi }}$$

The requirement that the matrix ensemble to be a gaussian ensemble is too strong for the Wigner semicircle law. Indeed, the theorem applies generally for much more generic matrix ensembles.

As Coulomb gas

The joint density $\rho _{\beta ,N}$ can be written as a Gibbs measure:$$\rho _{\beta ,N}={\frac {1}{Z_{\beta ,N}^{\text{CG}}}}e^{-\beta E_{N}}$$with the energy function (also called the Hamiltonian) $E_{N}={\frac {1}{4}}\sum _{i=1}^{N}\lambda _{i}^{2}-\sum _{1\leq i<j\leq N}\ln |\lambda _{i}-\lambda _{j}|$. This can be interpreted physically as a Boltzmann distribution of a physical system consisting of $N$ identical unit electric charges constrained to move on the real line, repelling each other via the two-dimensional Coulomb potential $-\ln |x-y|$, while being attracted to the origin via a quadratic potential ${\frac {1}{4}}x^{2}$. This is the Coulomb gas model for the eigenvalues.

In the macroscopic limit, one rescales $\lambda _{i}=N^{1/2}x_{i}$ and defines the empirical measure $\mu _{N}=N^{-1}\sum _{i=1}^{N}\delta _{x_{i}}$, obtaining $E_{N}\approx {\frac {1}{2}}N^{2}\left({\mathcal {E}}[\mu ]+{\frac {1}{2}}\ln N\right)$, where the mean-field functional ${\mathcal {E}}[\mu ]={\frac {1}{2}}\int x^{2}\mu (dx)-\iint \ln |x-y|\mu (dx)\mu (dy)$.

The minimizer of ${\mathcal {E}}$ over probability measures is the Wigner semicircle law $d\mu _{sc}(x)=(2\pi )^{-1}{\sqrt {4-x^{2}}}1_{\{|x|\leq 2\}}dx$, which gives the limiting eigenvalue density. The value ${\mathcal {E}}[\mu _{sc}]$ yields the leading order $N^{2}$ term in $\ln Z_{\beta ,N}$, termed the Coulomb gas free energy.

Alternatively, suppose that there exists a $\rho _{b}$, such that the quadratic electric potential can be recreated (up to an additive constant) via$$\int _{-2{\sqrt {N}}}^{2{\sqrt {N}}}-\ln |x-y|\rho _{b}(y)dy={\frac {1}{4}}x^{2}+C,\quad x\in [-2{\sqrt {N}},2{\sqrt {N}}].$$Then, imposing a fixed background negative electric charge of density $|\rho _{b}(y)|$ exactly cancels out the electric repulsion between the freely moving positive charges. Such a function does exist: $\rho _{b}(y)=-{\frac {\sqrt {4N-y^{2}}}{2N\pi }}$, which can be found by solving an integral equation. This indicates that the Wigner semicircle distribution is the equilibrium distribution.

Gaussian fluctuations about $\mu _{sc}$ obtained by expanding $E_{N}$ to second order produce the sine kernel in the bulk and the Airy kernel at the soft edge after proper rescaling.

Extreme value

The largest eigenvalue for GOE and GUE follows the Tracy–Widom distribution after proper centering and scaling.

Level spacings

From ordered eigenvalues $\lambda _{1}<\dots <\lambda _{n}<\lambda _{n+1}<\dots <\lambda _{N}$, define normalized spacings $s_{n}={\frac {\lambda _{n+1}-\lambda _{n}}{\langle s\rangle }}$ with mean spacing $\langle s\rangle$. This normalizes the spacings by:$$\int _{0}^{\infty }p_{\beta }(s)\,ds=1,\qquad \int _{0}^{\infty }s\,p_{\beta }(s)\,ds=1,\qquad \beta =1,2,4.$$With this, the approximate spacing distributions are $$p_{\beta }(s)={\begin{cases}{\frac {\pi }{2}}s\exp \left(-{\frac {\pi }{4}}s^{2}\right)&\beta =1\\{\frac {32}{\pi ^{2}}}s^{2}\exp \left(-{\frac {4}{\pi }}s^{2}\right)&\beta =2\\{\frac {2^{18}}{3^{6}\pi ^{3}}}s^{4}\exp \left(-{\frac {64}{9\pi }}s^{2}\right)&\beta =4\\\end{cases}}$$

Other properties

Point correlation

Matrix elements satisfy $\langle W_{ij}\rangle =0$ and $$\langle W_{ij}W_{mn}^{*}\rangle =\langle W_{ij}W_{nm}\rangle ={\frac {1}{N}}\delta _{im}\delta _{jn}+{\frac {2-\beta }{N\beta }}\delta _{in}\delta _{jm}$$with higher correlations given by Isserlis' theorem.

Moment generating functions

For GOE, $$\operatorname {E} \!\left[e^{\operatorname {Tr} (VW_{N})}\right]=\exp \left({\frac {1}{4N}}\|V+V^{\text{T}}\|_{F}^{2}\right)$$, where $\|\cdot \|_{F}$ is the Frobenius norm.

References

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• Mehta, M.L. (2004). Random Matrices. Amsterdam: Elsevier/Academic Press. ISBN 0-12-088409-7.
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• Livan, Giacomo; Novaes, Marcel; Vivo, Pierpaolo (2018). Introduction to Random Matrices. SpringerBriefs in Mathematical Physics. Vol. 26. Cham: Springer International Publishing. arXiv:1712.07903. doi:10.1007/978-3-319-70885-0. ISBN 978-3-319-70883-6.
• Potters, Marc; Bouchaud, Jean-Philippe (2020-11-30). A First Course in Random Matrix Theory: for Physicists, Engineers and Data Scientists. Cambridge University Press. doi:10.1017/9781108768900. ISBN 978-1-108-76890-0.