In probability theory, more specifically the study of random matrices, the circular law concerns the distribution of eigenvalues of an n × n {\displaystyle n\times n} random matrix with independent and identically distributed entries in the limit
n → ∞ {\displaystyle n\to \infty } . It asserts that for any sequence of random n × n matrices whose entries are independent and identically distributed random variables, all with mean zero and variance equal to 1/n, the limiting spectral distribution is the uniform distribution over the unit disc.
Ginibre ensembles Let G N {\displaystyle G_{N}} be a Ginibre ensemble matrix of size N × N {\displaystyle N\times N} . The real Ginibre ensemble is defined by sampling each entry IID form the standard normal distribution. That is, we have G i j ∼ N ( 0 , 1 ) {\displaystyle G_{ij}\sim {\mathcal {N}}\left(0,1\right)} . The complex Ginibre ensemble is defined as G i j ∼ N ( 0 , 1 2 ) + i N ( 0 , 1 2 ) {\displaystyle G_{ij}\sim {\mathcal {N}}\left(0,{\frac {1}{2}}\right)+i{\mathcal {N}}\left(0,{\frac {1}{2}}\right)} . The quaternionic Ginibre ensemble is defined as G i j ∼ N ( 0 , 1 4 ) + i N ( 0 , 1 4 ) + j N ( 0 , 1 4 ) + k N ( 0 , 1 4 ) {\displaystyle G_{ij}\sim {\mathcal {N}}\left(0,{\frac {1}{4}}\right)+i{\mathcal {N}}\left(0,{\frac {1}{4}}\right)+j{\mathcal {N}}\left(0,{\frac {1}{4}}\right)+k{\mathcal {N}}\left(0,{\frac {1}{4}}\right)} . Although, since the quaternion number system is inconvenient, it is usually not sampled as a quaternion matrix of shape n × n {\displaystyle n\times n} , but rather as a complex matrix of shape 2 N × 2 N {\displaystyle 2N\times 2N} , divided into 2 × 2 {\displaystyle 2\times 2} blocks of form [ z w − w ¯ z ¯ ] {\displaystyle {\begin{bmatrix}z&w\\-{\bar {w}}&{\bar {z}}\end{bmatrix}}} , such that each z , w {\displaystyle z,w} is IID sampled from N ( 0 , 1 4 ) + i N ( 0 , 1 4 ) {\displaystyle {\mathcal {N}}\left(0,{\frac {1}{4}}\right)+i{\mathcal {N}}\left(0,{\frac {1}{4}}\right)} . The probability measure of the Ginibre ensemble satisfies ln ρ ( G ) = − β 2 ∑ i j | G i j | 2 − ln Z β , N = β 2 Tr ( G G ∗ ) − ln Z β , N {\displaystyle \ln \rho (G)=-{\frac {\beta }{2}}\sum _{ij}|G_{ij}|^{2}-\ln Z_{\beta ,N}={\frac {\beta }{2}}\operatorname {Tr} (GG^{*})-\ln Z_{\beta ,N}} where
β = 1 , 2 , 4 {\displaystyle \beta =1,2,4} respectively for the real, complex, and quaternionic cases;
Z β , N {\displaystyle Z_{\beta ,N}} is a normalization factor;
Tr {\displaystyle \operatorname {Tr} } is the trace;
∗ {\displaystyle ^{*}} is the matrix adjoint. The most commonly used case is β = 2 {\displaystyle \beta =2} , and when "Ginibre ensemble" is spoken of, it by default means the β = 2 {\displaystyle \beta =2} case. By analogy with the gaussian ensembles, the cases of β = 1 , 2 , 4 {\displaystyle \beta =1,2,4} are also called the GinOE, GinUE, GinSE, meaning "Ginibre Orthogonal/Unitary/Symplectic Ensemble".
Spectral distribution
Probability density function For β = 2 {\displaystyle \beta =2} , the eigenvalues of G N {\displaystyle G_{N}} are distributed according to ρ N ( z 1 , … , z N ) = 1 Z exp ( − ∑ k = 1 N | z k | 2 ) ∏ 1 ≤ j < k ≤ N | z j − z k | 2 {\displaystyle \rho _{N}\left(z_{1},\ldots ,z_{N}\right)={\frac {1}{Z}}\exp \left(-\sum _{k=1}^{N}\left|z_{k}\right|^{2}\right)\prod _{1\leq j<k\leq N}\left|z_{j}-z_{k}\right|^{2}} where Z = π N ∏ k = 1 N k ! {\displaystyle Z=\pi ^{N}\prod _{k=1}^{N}k!} is a Selberg integral. Ignoring the term Z {\displaystyle Z} , the rest of the formula can be obtained by exploiting the biunitary symmetry of the ensemble. That is, for any unitary U , V {\displaystyle U,V} , the ensemble U G V {\displaystyle UGV} has the same distribution. For β = 4 {\displaystyle \beta =4} , the 2 N × 2 N {\displaystyle 2N\times 2N} matrix has complex eigenvalues that come in conjugate pairs. Index the eigenvalues as z 1 , … , z n , z n + 1 , … , z 2 n {\displaystyle z_{1},\dots ,z_{n},z_{n+1},\dots ,z_{2n}} such that z j = z ¯ n + j {\displaystyle z_{j}={\bar {z}}_{n+j}} , then ρ N ( z 1 , … , z N ) ∝ ∏ l = 1 N e − 2 | z l | 2 | z l − z ¯ l | 2 ∏ 1 ≤ j < k ≤ N | z k − z j | 2 | z k − z ¯ j | 2 , Im z l > 0 {\displaystyle \rho _{N}\left(z_{1},\ldots ,z_{N}\right)\propto \prod _{l=1}^{N}e^{-2\left|z_{l}\right|^{2}}\left|z_{l}-{\bar {z}}_{l}\right|^{2}\prod _{1\leq j<k\leq N}\left|z_{k}-z_{j}\right|^{2}\left|z_{k}-{\bar {z}}_{j}\right|^{2},\quad \operatorname {Im} z_{l}>0} For β = 1 {\displaystyle \beta =1} , the N × N {\displaystyle N\times N} matrix has N {\displaystyle N} complex eigenvalues, and each eigenvalue has a conjugate that is also an eigenvalue. However, they may no longer come in conjugate pairs, since some eigenvalues may be purely real. It is not even absolutely continuous, thus does not have a probability density function, but decomposes into sectors depending on the number of real eigenvalues. However, at the N → ∞ {\displaystyle N\to \infty } limit, the circular law is recovered, since there are only O ( N ) {\displaystyle O({\sqrt {N}})} eigenvalues exactly on the real line.
Determinantal point process For β = 2 {\displaystyle \beta =2} , the eigenvalues make up a determinantal point process ρ ( k ) , N ( z 1 , … , z k ) = det [ K N ( z j , z l ) ] j , l = 1 k {\displaystyle \rho _{(k),N}\left(z_{1},\ldots ,z_{k}\right)=\det \left[K_{N}\left(z_{j},z_{l}\right)\right]_{j,l=1}^{k}} with correlation kernel K N ( w , z ) = 1 π e − ( | w | 2 + | z | 2 ) / 2 ∑ j = 1 N ( w z ¯ ) j − 1 ( j − 1 ) ! = 1 π e − ( | w | 2 + | z | 2 ) / 2 e w z ¯ Γ ( N ; w z ¯ ) Γ ( N ) {\displaystyle K_{N}(w,z)={\frac {1}{\pi }}e^{-\left(|w|^{2}+|z|^{2}\right)/2}\sum _{j=1}^{N}{\frac {(w{\bar {z}})^{j-1}}{(j-1)!}}={\frac {1}{\pi }}e^{-\left(|w|^{2}+|z|^{2}\right)/2}e^{w{\bar {z}}}{\frac {\Gamma (N;w{\bar {z}})}{\Gamma (N)}}} where Γ ( j ; x ) = ∫ x ∞ t j − 1 e − t d t {\displaystyle \Gamma (j;x)=\int _{x}^{\infty }t^{j-1}e^{-t}dt} denotes the upper incomplete gamma function. It has the following asymptotics K ∞ b ( w , z ) := lim N → ∞ K N ( w , z ) = 1 π e − ( | w | 2 + | z | 2 ) / 2 e w z ¯ , {\displaystyle K_{\infty }^{\mathrm {b} }(w,z):=\lim _{N\rightarrow \infty }K_{N}(w,z)={\frac {1}{\pi }}e^{-\left(|w|^{2}+|z|^{2}\right)/2}e^{w{\bar {z}}},}
K ∞ e ( z 1 , z 2 ) := lim N → ∞ K N ( − i N + z 1 , − i N + z 2 ) = e − ( | z 1 | 2 + | z 2 | 2 ) / 2 e z 1 z ¯ 2 h ( 1 2 ( − i z 1 + i z ¯ 2 ) ) {\displaystyle K_{\infty }^{\mathrm {e} }\left(z_{1},z_{2}\right):=\lim _{N\rightarrow \infty }K_{N}\left(-i{\sqrt {N}}+z_{1},-i{\sqrt {N}}+z_{2}\right)=e^{-\left(\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}\right)/2}e^{z_{1}{\bar {z}}_{2}}h\left({\frac {1}{2}}\left(-iz_{1}+i{\bar {z}}_{2}\right)\right)} where h ( z ) = 1 2 π ( 1 + erf ( 2 z ) ) {\displaystyle h(z)={\frac {1}{2\pi }}(1+\operatorname {erf} ({\sqrt {2}}z))} .
Global law
Plugging in the correlation kernel, the average distribution of all eigenvalues is 1 N K N ( z , z ) = Γ ( N ; | z | 2 ) π N ! = ∫ | z | 2 ∞ t N − 1 e − t d t π N ! {\displaystyle {\frac {1}{N}}K_{N}(z,z)={\frac {\Gamma (N;|z|^{2})}{\pi N!}}={\frac {\int _{|z|^{2}}^{\infty }t^{N-1}e^{-t}dt}{\pi N!}}} Scaling down by N {\displaystyle {\sqrt {N}}} , we find that the average distribution of the eigenvalues of 1 N G N {\textstyle {\frac {1}{\sqrt {N}}}G_{N}} to have probability density function ρ ( z ) = N N π ( N − 1 ) ! ∫ | z | 2 ∞ t N − 1 e − N t d t {\displaystyle \rho (z)={\frac {N^{N}}{\pi (N-1)!}}\int _{|z|^{2}}^{\infty }t^{N-1}e^{-Nt}dt} which rapidly converges to { 1 π if | z | < 1 0 if | z | > 1 {\displaystyle {\begin{cases}{\frac {1}{\pi }}&{\text{if }}|z|<1\\0&{\text{if }}|z|>1\end{cases}}} .More strongly, we have the strong global law. Let ( G N ) N = 1 ∞ {\displaystyle (G_{N})_{N=1}^{\infty }} be a sequence sampled from the complex Ginibre ensemble. Define μ 1 n G n {\displaystyle \displaystyle \mu _{{\frac {1}{\sqrt {n}}}G_{n}}} to be the empirical spectral measure of 1 N G N {\textstyle {\frac {1}{\sqrt {N}}}G_{N}} . Then, almost surely (i.e. with probability one), the sequence of measures converges in distribution to the uniform measure on the unit disk.
As a Coulomb gas Recall the spectral distribution ρ N ( z 1 , … , z N ) = 1 Z exp ( − ∑ k = 1 N | z k | 2 ) ∏ 1 ≤ j < k ≤ N | z j − z k | 2 {\displaystyle \rho _{N}\left(z_{1},\ldots ,z_{N}\right)={\frac {1}{Z}}\exp \left(-\sum _{k=1}^{N}\left|z_{k}\right|^{2}\right)\prod _{1\leq j<k\leq N}\left|z_{j}-z_{k}\right|^{2}} It can be interpreted as the Boltzmann distribution for a Coulomb gas, or more specifically a two-dimensional one-component plasma (OCP), at inverse temperature β = 2 {\displaystyle \beta =2} . Note that here β {\displaystyle \beta } is used to mean something different, and may take any value within ( 0 , ∞ ) {\displaystyle (0,\infty )} . The gas contains N {\displaystyle N} identical particles, all placed within the plane C {\displaystyle \mathbb {C} } , with total energy E = 1 2 ∑ k = 1 N | z k | 2 − ∑ 1 ≤ j < k ≤ N ln | z j − z k | {\displaystyle E={\frac {1}{2}}\sum _{k=1}^{N}|z_{k}|^{2}-\sum _{1\leq j<k\leq N}\ln |z_{j}-z_{k}|} The first term indicates that every particle is attracted to the origin by a force of magnitude F k = | z k | {\displaystyle F_{k}=|z_{k}|} . The second term indicates that every particle pair is repelling each other by a force of magnitude F j k = 1 | z j − z k | {\displaystyle F_{jk}={\frac {1}{|z_{j}-z_{k}|}}} . For general inverse temperature β {\displaystyle \beta } , the OCP has partition function Z N OCP ( β ) = ∫ C N e − β E d N z {\displaystyle Z_{N}^{\text{OCP}}(\beta )=\int _{\mathbb {C} ^{N}}e^{-\beta E}d^{N}z} , and free energy − β − 1 ln Z β , N OCP {\displaystyle -\beta ^{-1}\ln Z_{\beta ,N}^{\text{OCP}}} . However, it is theoretically more natural to consider the normalized partition function Z N D R , O C P ( β ) = 1 N ! A N , β Z N O C P ( β ) , A N , β = e − β N 2 ( 1 4 log N − 3 8 ) {\displaystyle Z_{N}^{D_{R},\mathrm {OCP} }(\beta )={\frac {1}{N!}}A_{N,\beta }Z_{N}^{\mathrm {OCP} }(\beta ),\quad A_{N,\beta }=e^{-\beta N^{2}\left({\frac {1}{4}}\log N-{\frac {3}{8}}\right)}} where the N ! {\displaystyle N!} part accounts for the fact that the particles are indistinguishable from each other, and A N , β {\displaystyle A_{N,\beta }} removes the self-energy of the average plasma, that is, the self-energy of a disk of radius R = N {\displaystyle R={\sqrt {N}}} and charge density 1 π {\displaystyle {\frac {1}{\pi }}} . Thus, Z N D R , O C P ( β ) {\displaystyle Z_{N}^{D_{R},\mathrm {OCP} }(\beta )} is the partition function of a "charge neutral" OCP. The log-partition function satisfies − ln