In random matrix theory, the Gaussian ensembles are specific probability distributions over self-adjoint matrices whose entries are independently sampled from the gaussian distribution. They are among the most-commonly studied matrix ensembles, fundamental to both mathematics and physics. The three main examples are the Gaussian orthogonal (GOE), unitary (GUE), and symplectic (GSE) ensembles. These are classified by the Dyson index β, which takes values 1, 2, and 4 respectively, counting the number of real components per matrix element (1 for real elements, 2 for complex elements, 4 for quaternions). The index can be extended to take any real positive value. The gaussian ensembles are also called the Wigner ensembles, or the Hermite ensembles.
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.
β {\displaystyle \beta } : a positive real number. Called the Dyson index. The cases of β = 1 , 2 , 4 {\displaystyle \beta =1,2,4} are special.
N {\displaystyle N} : the side-length of a matrix. Always a positive integer.
W N {\displaystyle W_{N}} : a matrix sampled from a Gaussian ensemble with size N × N {\displaystyle N\times N} . The letter W {\displaystyle W} stands for "Wigner".
M ∗ {\displaystyle M^{*}} : the adjoint of a matrix. We assume W N = W N ∗ {\displaystyle W_{N}=W_{N}^{*}} (self-adjoint) when W N {\displaystyle W_{N}} is sampled from a gaussian ensemble. If M {\displaystyle M} is real, then M ∗ {\displaystyle M^{*}} is its transpose. If M {\displaystyle M} is complex or quaternionic, then M ∗ {\displaystyle M^{*}} is its conjugate transpose.
λ 1 , … , λ N {\displaystyle \lambda _{1},\dots ,\lambda _{N}} : the eigenvalues of the matrix, which are all real, since the matrices are always assumed to be self-adjoint.
σ d 2 {\displaystyle \sigma _{d}^{2}} : the variance of on-diagonal matrix entries. We assume that for each N {\displaystyle N} , all on-diagonal matrix entries have the same variance. It is always defined as E [ | W N , | 2 ] {\displaystyle \mathbb {E} [|W_{N,}|^{2}]} .
σ o d 2 {\displaystyle \sigma _{od}^{2}} : the variance of off-diagonal matrix entries. We assume that for each N {\displaystyle N} , all off-diagonal matrix entries have the same variance. It is always defined as E [ | W N , i j | 2 ] {\displaystyle \mathbb {E} [|W_{N,ij}|^{2}]} where i ≠ j {\displaystyle i\neq j} . For a complex number, | a + b i | 2 = a 2 + b 2 {\displaystyle |a+bi|^{2}=a^{2}+b^{2}} . For a quaternion, | a + b i + c j + d k | 2 = a 2 + b 2 + c 2 + d 2 {\displaystyle |a+bi+cj+dk|^{2}=a^{2}+b^{2}+c^{2}+d^{2}} .
Z {\displaystyle Z} : the partition function.
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.
There are equivalent definitions for the GβE(N) ensembles, given below.
By sampling For all β = 1 , 2 , 4 {\displaystyle \beta =1,2,4} cases, the GβE(N) ensemble is defined by how it is sampled:
Sample a gaussian matrix X N {\displaystyle X_{N}} , such that all its entries are IID sampled from the corresponding standard normal distribution. If β = 1 {\displaystyle \beta =1} , then X N , i j ∼ N ( 0 , 1 ) {\displaystyle X_{N,ij}\sim {\mathcal {N}}(0,1)} . If β = 2 {\displaystyle \beta =2} , then X N , i j ∼ N ( 0 , 1 / 2 ) + i N ( 0 , 1 / 2 ) {\displaystyle X_{N,ij}\sim {\mathcal {N}}(0,1/2)+i{\mathcal {N}}(0,1/2)} . If β = 4 {\displaystyle \beta =4} , then X N , i j ∼ N ( 0 , 1 / 4 ) + i N ( 0 , 1 / 4 ) + j N ( 0 , 1 / 4 ) + k N ( 0 , 1 / 4 ) {\displaystyle 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 = 1 2 ( X + X ∗ ) {\displaystyle W_{N}={\frac {1}{\sqrt {2}}}(X+X^{*})} .
By density For all β = 1 , 2 , 4 {\displaystyle \beta =1,2,4} cases, the GβE(N) ensemble is defined with density function ρ ( W N ) = 1 Z e − β 4 ∑ i = 1 N W N , i i 2 − β 2 ∑ 1 ≤ i < j ≤ N | W N , i j | 2 = 1 Z e − β 4 T r W N 2 {\displaystyle \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 1 2 N ( 2 π β ) 1 2 N + 1 4 β N ( N − 1 ) {\displaystyle 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 × N {\displaystyle N\times N} symmetric matrices with density function ρ ( W N ) = 1 Z e − 1 4 ∑ i = 1 N W N , i i 2 − 1 2 ∑ 1 ≤ i < j ≤ N W N , i j 2 = 1 Z e − 1 4 T r W N 2 {\displaystyle \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 1 4 N ( N + 3 ) π 1 4 N ( N + 1 ) {\displaystyle Z=2^{{\frac {1}{4}}N(N+3)}\pi ^{{\frac {1}{4}}N(N+1)}} . Explicitly, since there are only 1 2 N ( N + 1 ) {\displaystyle {\frac {1}{2}}N(N+1)} degrees of freedom, the parameterization is as follows: ρ ( W N ) ∏ 1 ≤ i ≤ j ≤ N d W N , i j {\displaystyle \rho (W_{N})\prod _{1\leq i\leq j\leq N}dW_{N,ij}} where we pick the upper diagonal entries { W i j } 1 ≤ i ≤ j ≤ N {\displaystyle \{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 × N {\displaystyle N\times N} Hermitian matrices with density function ρ ( W N ) = 1 Z e − 1 2 ∑ i = 1 N W N , i i 2 − ∑ 1 ≤ i < j ≤ N | W N , i j | 2 = 1 Z e − 1 2 T r W N 2 . {\displaystyle \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 1 2 N π 1 2 N 2 {\displaystyle Z=2^{{\frac {1}{2}}N}\pi ^{{\frac {1}{2}}N^{2}}} . Explicitly, since there are only N 2 {\displaystyle N^{2}} degrees of freedom, the parameterization is as follows:
ρ ( W N ) ∏ i = 1 N d W N , i i ∏ 1 ≤ i < j ≤ N d ( R e W N , i j ) d ( I m W N , i j ) {\displaystyle \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 , i i } 1 ≤ i ≤ N ∪ { R e W N , i j , I m W N , i j } 1 ≤ i < j ≤ N {\displaystyle \{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 × N {\displaystyle N\times N} self‑adjoint quaternionic matrices with density function ρ ( W N ) = 1 Z e − ∑ i = 1 N W N , i i 2 − 2 ∑ 1 ≤ i < j ≤ N | W N , i j | 2 = 1 Z e − T r W N 2 . {\displaystyle \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 ) π 1 2 N ( 2 N − 1 ) {\displaystyle Z=2^{-N(N-1)}\pi ^{{\frac {1}{2}}N(2N-1)}} . Explicitly, since there are only N ( 2 N − 1 ) {\displaystyle N(2N-1)} degrees of freedom, the parameterization is as follows: ρ ( W N ) ∏ i = 1 N d W N , i i ∏ 1 ≤ i < j ≤ N ∏ a = 0 3 d W N , i j ( a ) {\displaystyle \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 , i j = W N , i j ( 0 ) + i W N , i j ( 1 ) + j W N , i j ( 2 ) + k W N , i j ( 3 ) {\displaystyle 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 , i i } 1 ≤ i ≤ N ∪ { W N , i j ( a ) } 1 ≤ i < j ≤ N , 0 ≤ a ≤ 3 {\displaystyle \{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 β = 1 , 2 , 4 {\displaystyle \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 × N {\displaystyle N\times N} symmetric matrices satisfying the following properties:
Invariance under orthogonal transformation: For any fixed (not random) N × N {\displaystyle N\times N} orthogonal matrix O {\displaystyle O} , let M {\displaystyle M} be a random sample from the distribution. Then O M O T {\displaystyle OMO^{T}} has the same distribution as M {\displaystyle M} . Independence: The entries { M i j } 1 ≤ i ≤ j ≤ N {\displaystyle \{M_{ij}\}_{1\leq i\leq j\leq N}} are independently distributed. For GUE, consider a probability distribution over N × N {\displaystyle N\times N} Hermitian matrices satisfying the following properties:
Invariance under unitary transformation: For any fixed (not random) N × N {\displaystyle N\times N} unitary matrix U {\displaystyle U} , let M {\displaystyle M} be a random sample from the distribution. Then U M U ∗ {\displaystyle UMU^{*}} has the same distribution as M {\displaystyle M} . Independence: The entries { M i j } 1 ≤ i ≤ j ≤ N {\displaystyle \{M_{ij}\}_{1\leq i\leq j\leq N}} are independently distributed. For GSE, consider a probability distribution over N × N {\displaystyle N\times N} self-adjoint quaternionic matrices satisfying the following properties:
Invariance under symplectic transformation: For any fixed (not random) N × N {\displaystyle N\times N} symplectic matrix S {\displaystyle S} , let M {\displaystyle M} be a random sample from the distribution. Then S M S ∗ {\displaystyle SMS^{*}} has the same distribution as M {\displaystyle M} . Independence: The entries { M i j } 1 ≤ i ≤ j ≤ N {\displaystyle \{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 ρ ( M ) = 1 Z e − a Tr ( M 2 ) + b Tr ( M ) {\displaystyle \rho (M)={\frac {1}{Z}}e^{-a\operatorname {Tr} (M^{2})+b\operatorname {Tr} (M)}} , where a > 0 {\displaystyle a>0} and b , Z ∈ R {\displaystyle b,Z\in \mathbb {R} } . Thus, with the further specification of 1 N E [ Tr ( M ) ] = 0 , 1 N 2 E [ Tr ( M 2 ) ] = 1 + 2 / β − 1 N {\displaystyle {\frac {1}{N}}\mathbb {E} [\operatorname {Tr} (M)]=0,{\frac {1}{N^{2}}}\mathbb {E} [\operatorname {Tr} (M^{2})]=1+{\frac {2/\beta -1}{N}}} , we recover the GOE, GUE, GSE. Notably, if mere invariance is demanded, then any spectral distribution can be produced by multiplying with a function of form f ( Tr ( X ) , Tr ( X 2 ) , Tr ( X 3 ) , … ) {\displaystyle f(\operatorname {Tr} (X),\operatorname {Tr} (X^{2}),\operatorname {Tr} (X^{3}),\dots )} . More succinctly stated, each of GOE, GUE, GSE is uniquely specified by invariance, independence, the mean, and the variance.
By spectral distribution For all β = 1 , 2 , 4 {\displaystyle \beta =1,2,4} cases, the GβE(N) ensemble is defined as the ensemble obtained by A D A ∗ {\displaystyle ADA^{*}} , where
D = diag ( λ 1 , … , λ N ) {\displaystyle 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 {\displaystyle A} is an orthogonal/unitary/symplectic matrix sampled uniformly, that is, from the normalized Haar measure of the orthogonal/unitary/symplectic group. In this way, the GβE(N) ensemble may be defined after the spectral density is defined first, so that any method to motivate the spectral density then motivates the GβE(N) ensemble, and vice versa.
By maximal entropy For all β = 1 , 2 , 4 {\displaystyle \beta =1,2,4} cases, the GβE(N) ensemble is uniquely characterized as the absolutely continuous probability distribution ρ {\displaystyle \rho } over N × N {\displaystyle N\times N} real/complex/quaternionic symmetric/orthogonal/symplectic matrices that maximizes entropy E M ∼ ρ [ − ln ρ ( M ) ] {\displaystyle \mathbb {E} _{M\sim \rho }[-\ln \rho (M)]} , under the constraint of 1 N 2 E M ∼ ρ [ Tr ( M 2 ) ] = 1 + 2 / β − 1 N {\displaystyle {\frac {1}{N^{2}}}\mathbb {E} _{M\sim \rho }[\operatorname {Tr} (M^{2})]=1+{\frac {2/\beta -1}{N}}} .
Spectral density For eigenvalues λ 1 , … , λ N {\displaystyle \lambda _{1},\dots ,\lambda _{N}} the joint density of GβE(N) is ρ β , N ( λ 1 , … , λ N ) = 1 Z β , N e − β 4 ∑ i = 1 N λ i 2 ∏ 1 ≤ i < j ≤ N | λ i − λ j | β = 1 Z β , N e − β 4 ‖ λ ‖ 2 2 | Δ N ( λ ) | β {\displaystyle \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 Δ N {\displaystyle \Delta _{N}} is the Vandermonde determinant, and the partition function Z β , N {\displaystyle Z_{\beta ,N}} is explicitly evaluated as a Selberg integral: Z β , N = ∫ R N e − β 4 ∑ i = 1 N λ