Random matrix

Inprobability theoryandmathematical physics,arandom matrixis amatrix-valuedrandom variable—that is, a matrix in which some or all of its entries aresampledrandomly from aprobability distribution.Random matrix theory (RMT)is the study of properties of random matrices, often as they become large. RMT provides techniques likemean-field theory,diagrammatic methods, thecavity method,or thereplica methodto compute quantities liketraces,spectral densities,or scalar products between eigenvectors. Many physical phenomena, such as thespectrumofnucleiof heavy atoms,thethermal conductivityof alattice,or the emergence ofquantum chaos,can be modeled mathematically as problems concerning large, random matrices.

Applications

Physics

Innuclear physics,random matrices were introduced byEugene Wignerto model the nuclei of heavy atoms.Wigner postulated that the spacings between the lines in the spectrum of a heavy atom nucleus should resemble the spacings between theeigenvaluesof a random matrix, and should depend only on the symmetry class of the underlying evolution.Insolid-state physics,random matrices model the behaviour of large disorderedHamiltoniansin themean-field approximation.

Inquantum chaos,the Bohigas–Giannoni–Schmit (BGS) conjecture asserts that the spectral statistics of quantum systems whose classical counterparts exhibit chaotic behaviour are described by random matrix theory.

Inquantum optics,transformations described by random unitary matrices are crucial for demonstrating the advantage of quantum over classical computation (see, e.g., theboson samplingmodel).Moreover, such random unitary transformations can be directly implemented in an optical circuit, by mapping their parameters to optical circuit components (that isbeam splittersand phase shifters).

Random matrix theory has also found applications to the chiral Dirac operator inquantum chromodynamics,quantum gravityin two dimensions,mesoscopic physics,spin-transfer torque,thefractional quantum Hall effect,Anderson localization,quantum dots,andsuperconductors

Mathematical statistics and numerical analysis

Inmultivariate statistics,random matrices were introduced byJohn Wishart,who sought toestimate covariance matricesof large samples.Chernoff-,Bernstein-, andHoeffding-type inequalities can typically be strengthened when applied to the maximal eigenvalue (i.e. the eigenvalue of largest magnitude) of a finite sum of randomHermitian matrices.Random matrix theory is used to study the spectral properties of random matrices—such as sample covariance matrices—which is of particular interest inhigh-dimensional statistics.Random matrix theory also saw applications inneuronal networksanddeep learning,with recent work utilizing random matrices to show that hyper-parameter tunings can be cheaply transferred between large neural networks without the need for re-training.

Innumerical analysis,random matrices have been used since the work ofJohn von NeumannandHerman Goldstineto describe computation errors in operations such asmatrix multiplication.Although random entries are traditional "generic" inputs to an algorithm, theconcentration of measureassociated with random matrix distributions implies that random matrices will not test large portions of an algorithm's input space.

Number theory

Innumber theory,the distribution of zeros of theRiemann zeta function(and otherL-functions) is modeled by the distribution of eigenvalues of certain random matrices.The connection was first discovered byHugh MontgomeryandFreeman Dyson.It is connected to theHilbert–Pólya conjecture.

Free probability

The relation offree probabilitywith random matricesis a key reason for the wide use of free probability in other subjects. Voiculescu introduced the concept of freeness around 1983 in an operator algebraic context; at the beginning there was no relation at all with random matrices. This connection was only revealed later in 1991 by Voiculescu;he was motivated by the fact that the limit distribution which he found in his free central limit theorem had appeared before in Wigner's semi-circle law in the random matrix context.

Computational neuroscience

In the field of computational neuroscience, random matrices are increasingly used to model the network of synaptic connections between neurons in the brain. Dynamical models of neuronal networks with random connectivity matrix were shown to exhibit a phase transition to chaoswhen the variance of the synaptic weights crosses a critical value, at the limit of infinite system size. Results on random matrices have also shown that the dynamics of random-matrix models are insensitive to mean connection strength. Instead, the stability of fluctuations depends on connection strength variationand time to synchrony depends on network topology.

In the analysis of massive data such asfMRI,random matrix theory has been applied in order to perform dimension reduction. When applying an algorithm such asPCA,it is important to be able to select the number of significant components. The criteria for selecting components can be multiple (based on explained variance, Kaiser's method, eigenvalue, etc.). Random matrix theory in this content has its representative theMarchenko-Pastur distribution,which guarantees the theoretical high and low limits of the eigenvalues associated with a random variable covariance matrix. This matrix calculated in this way becomes the null hypothesis that allows one to find the eigenvalues (and their eigenvectors) that deviate from the theoretical random range. The components thus excluded become the reduced dimensional space (see examples in fMRI).

Optimal control

Inoptimal controltheory, the evolution ofnstate variables through time depends at any time on their own values and on the values ofkcontrol variables. With linear evolution, matrices of coefficients appear in the state equation (equation of evolution). In some problems the values of the parameters in these matrices are not known with certainty, in which case there are random matrices in the state equation and the problem is known as one ofstochastic control.A key result in the case oflinear-quadratic controlwith stochastic matrices is that thecertainty equivalence principledoes not apply: while in the absence ofmultiplier uncertainty(that is, with only additive uncertainty) the optimal policy with a quadratic loss function coincides with what would be decided if the uncertainty were ignored, the optimal policy may differ if the state equation contains random coefficients.

Computational mechanics

Incomputational mechanics,epistemic uncertainties underlying the lack of knowledge about the physics of the modeled system give rise to mathematical operators associated with the computational model, which are deficient in a certain sense. Such operators lack certain properties linked to unmodeled physics. When such operators are discretized to perform computational simulations, their accuracy is limited by the missing physics. To compensate for this deficiency of the mathematical operator, it is not enough to make the model parameters random, it is necessary to consider a mathematical operator that is random and can thus generate families of computational models in the hope that one of these captures the missing physics. Random matrices have been used in this sense,with applications in vibroacoustics, wave propagations, materials science, fluid mechanics, heat transfer, etc.

Engineering

Random matrix theory can be applied to the electrical and communications engineering research efforts to study, model and develop Massive Multiple-Input Multiple-Output (MIMO) radio systems.

History

Random matrix theory first gained attention beyond mathematics literature in the context of nuclear physics. Experiments byEnrico Fermiand others demonstrated evidence that individualnucleonscannot be approximated to move independently, leadingNiels Bohrto formulate the idea of acompound nucleus.Because there was no knowledge of direct nucleon-nucleon interactions,Eugene WignerandLeonard Eisenbudapproximated that the nuclearHamiltoniancould be modeled as a random matrix. For larger atoms, the distribution of theenergy eigenvaluesof the Hamiltonian could be computed in order to approximatescattering cross sectionsby invoking theWishart distribution.

Gaussian ensembles

The most-commonly studied random matrixdistributionsare the Gaussian ensembles: GOE, GUE and GSE. They are often denoted by theirDysonindex,β= 1 for GOE,β= 2 for GUE, andβ= 4 for GSE. This index counts the number of real components per matrix element.

Definitions

TheGaussian unitary ensemble${\text{GUE}}(n)$is described by theGaussian measurewith density $${\frac {1}{Z_{{\text{GUE}}(n)}}}e^{-{\frac {n}{2}}\mathrm {tr} H^{2}}$$ on the space of$n\times n$Hermitian matrices$H=(H_{ij})_{i,j=1}^{n}$.Here $$Z_{{\text{GUE}}(n)}=2^{n/2}\left({\frac {\pi }{n}}\right)^{{\frac {1}{2}}n^{2}}$$ is a normalization constant, chosen so that the integral of the density is equal to one. The termunitaryrefers to the fact that the distribution is invariant under unitary conjugation. The Gaussian unitary ensemble modelsHamiltonianslacking time-reversal symmetry.

TheGaussian orthogonal ensemble${\text{GOE}}(n)$is described by the Gaussian measure with density $${\frac {1}{Z_{{\text{GOE}}(n)}}}e^{-{\frac {n}{4}}\mathrm {tr} H^{2}}$$ on the space ofn×nreal symmetric matricesH= (H_ij)ⁿ
_i,j=1.Its distribution is invariant under orthogonal conjugation, and it models Hamiltonians with time-reversal symmetry. Equivalently, it is generated by$H=(G+G^{T})/{\sqrt {2n}}$,where$G$is an$n\times n$matrix with IID samples from the standard normal distribution.

TheGaussian symplectic ensemble${\text{GSE}}(n)$is described by the Gaussian measure with density $${\frac {1}{Z_{{\text{GSE}}(n)}}}e^{-n\mathrm {tr} H^{2}}$$ on the space ofn×nHermitianquaternionic matrices,e.g. symmetric square matrices composed ofquaternions,H= (H_ij)ⁿ
_i,j=1.Its distribution is invariant under conjugation by thesymplectic group,and it models Hamiltonians with time-reversal symmetry but no rotational symmetry.

Point correlation functions

The ensembles as defined here have Gaussian distributed matrix elements with mean ⟨H_ij⟩ = 0, and two-point correlations given by $$\langle H_{ij}H_{mn}^{*}\rangle =\langle H_{ij}H_{nm}\rangle ={\frac {1}{n}}\delta _{im}\delta _{jn}+{\frac {2-\beta }{n\beta }}\delta _{in}\delta _{jm},$$ from which all higher correlations follow byIsserlis' theorem.

Moment generating functions

Themoment generating functionfor the GOE is$$E[e^{tr(VH)}]=e^{{\frac {1}{4N}}\|V+V^{T}\|_{F}^{2}}$$where$\|\cdot \|_{F}$is theFrobenius norm.

Spectral density

Spectral density of GOE/GUE/GSE, as$N=2^{0},2^{1},...,2^{5}$.They are normalized so that the distributions converge to thesemicircle distribution.The number of "humps" is equal to N.

The jointprobability densityfor theeigenvaluesλ₁,λ₂,...,λ_nof GUE/GOE/GSE is given by

$${\frac {1}{Z_{\beta,n}}}\prod _{k=1}^{n}e^{-{\frac {\beta }{4}}\lambda _{k}^{2}}\prod _{i<j}\left|\lambda _{j}-\lambda _{i}\right|^{\beta }~,$$(1)

whereZ_β,nis a normalization constant which can be explicitly computed, seeSelberg integral.In the case of GUE (β= 2), the formula (1) describes adeterminantal point process.Eigenvalues repel as the joint probability density has a zero (of$\beta$th order) for coinciding eigenvalues$\lambda _{j}=\lambda _{i}$.

The distribution of the largest eigenvalue for GOE, and GUE, are explicitly solvable.They converge to theTracy–Widom distributionafter shifting and scaling appropriately.

Convergence to Wigner semicircular distribution

The spectrum, divided by${\sqrt {N\sigma ^{2}}}$,converges in distribution to thesemicircular distributionon the interval$[-2,+2]$:$\rho (x)={\frac {1}{2\pi }}{\sqrt {4-x^{2}}}$.Here$\sigma ^{2}$is the variance of off-diagonal entries. The variance of the on-diagonal entries do not matter.

Distribution of level spacings

From the ordered sequence of eigenvalues$\lambda _{1}<\ldots <\lambda _{n}<\lambda _{n+1}<\ldots$,one defines the normalizedspacings$s=(\lambda _{n+1}-\lambda _{n})/\langle s\rangle$,where$\langle s\rangle =\langle \lambda _{n+1}-\lambda _{n}\rangle$is the mean spacing. The probability distribution of spacings is approximately given by, $$p_{1}(s)={\frac {\pi }{2}}s\,e^{-{\frac {\pi }{4}}s^{2}}$$ for the orthogonal ensemble GOE$\beta =1$, $$p_{2}(s)={\frac {32}{\pi ^{2}}}s^{2}\mathrm {e} ^{-{\frac {4}{\pi }}s^{2}}$$ for the unitary ensemble GUE$\beta =2$,and $$p_{4}(s)={\frac {2^{18}}{3^{6}\pi ^{3}}}s^{4}e^{-{\frac {64}{9\pi }}s^{2}}$$ for the symplectic ensemble GSE$\beta =4$.

The numerical constants are such that$p_{\beta }(s)$is normalized: $$\int _{0}^{\infty }ds\,p_{\beta }(s)=1$$ and the mean spacing is, $$\int _{0}^{\infty }ds\,s\,p_{\beta }(s)=1,$$ for$\beta =1,2,4$.

Generalizations

Wigner matricesare random Hermitian matrices${\textstyle H_{n}=(H_{n}(i,j))_{i,j=1}^{n}}$such that the entries $$\left\{H_{n}(i,j)~,\,1\leq i\leq j\leq n\right\}$$ above the main diagonal are independent random variables with zero mean and have identical second moments.

Invariant matrix ensemblesare random Hermitian matrices with density on the space of real symmetric/Hermitian/quaternionic Hermitian matrices, which is of the form${\textstyle {\frac {1}{Z_{n}}}e^{-nV(\mathrm {tr} (H))}~,}$where the functionVis called the potential.

The Gaussian ensembles are the only common special cases of these two classes of random matrices. This is a consequence of a theorem by Porter and Rosenzweig.

Spectral theory of random matrices

The spectral theory of random matrices studies the distribution of the eigenvalues as the size of the matrix goes to infinity.

Empirical spectral measure

Theempirical spectral measureμ_HofHis defined by$$\mu _{H}(A)={\frac {1}{n}}\,\#\left\{{\text{eigenvalues of }}H{\text{ in }}A\right\}=N_{1_{A},H},\quad A\subset \mathbb {R}.$$

Usually, the limit of$\mu _{H}$is a deterministic measure; this is a particular case ofself-averaging.Thecumulative distribution functionof the limiting measure is called theintegrated density of statesand is denotedN(λ). If the integrated density of states is differentiable, its derivative is called thedensity of statesand is denotedρ(λ).

Alternative expressions

$$\mu _{H}(A)={\frac {1}{n}}\sum _{i}\delta _{\lambda _{i}}$$

Types of convergence

Given a matrix ensemble, we say that its spectral measures convergeweaklyto$\rho$iff for any measurable set$A$,the ensemble-average converges:$$\lim _{n\to \infty }\mathbb {E} _{H}[\mu _{H}(A)]=\rho (A)$$Convergenceweakly almost surely:If we sample$H_{1},H_{2},H_{3},\dots$independently from the ensemble, then with probability 1,$$\lim _{n\to \infty }\mu _{H_{n}}(A)=\rho (A)$$for any measurable set$A$.

In another sense,weak almost sure convergence means that we sample$H_{1},H_{2},H_{3},\dots$,not independently, but by "growing" (astochastic process), then with probability 1,$\lim _{n\to \infty }\mu _{H_{n}}(A)=\rho (A)$for any measurable set$A$.

For example, we can "grow" a sequence of matrices from the Gaussian ensemble as follows:

• Sample an infinite doubly infinite sequence of standard random variables$\{G_{i,j}\}_{i,j=1,2,3,\dots }$.
• Define each$H_{n}=(G_{n}+G_{n}^{T})/{\sqrt {2n}}$where$G_{n}$is the matrix made of entries$\{G_{i,j}\}_{i,j=1,2,\dots,n}$.

Note that generic matrix ensembles do not allow us to grow, but most of the common ones, such as the three Gaussian ensembles, do allow us to grow.

Global regime

In theglobal regime,one is interested in the distribution of linear statistics of the form$N_{f,H}=n^{-1}{\text{tr}}f(H)$.

The limit of the empirical spectral measure for Wigner matrices was described byEugene Wigner;seeWigner semicircle distributionandWigner surmise.As far as sample covariance matrices are concerned, atheory was developed by Marčenko and Pastur.

The limit of the empirical spectral measure of invariant matrix ensembles is described by a certain integral equation which arises frompotential theory.

Fluctuations

For the linear statisticsN_f,H=n⁻¹Σf(λ_j),one is also interested in the fluctuations about ∫f(λ)dN(λ). For many classes of random matrices, a central limit theorem of the form $${\frac {N_{f,H}-\int f(\lambda )\,dN(\lambda )}{\sigma _{f,n}}}{\overset {D}{\longrightarrow }}N(0,1)$$ is known.

The variational problem for the unitary ensembles

Consider the measure

$\mathrm {d} \mu _{N}(\mu )={\frac {1}{{\widetilde {Z}}_{N}}}e^{-H_{N}(\lambda )}\mathrm {d} \lambda,\qquad H_{N}(\lambda )=-\sum \limits _{j\neq k}\ln |\lambda _{j}-\lambda _{k}|+N\sum \limits _{j=1}^{N}Q(\lambda _{j}),$

where$Q(M)$is the potential of the ensemble and let$\nu$be the empirical spectral measure.

We can rewrite$H_{N}(\lambda )$with$\nu$as

$H_{N}(\lambda )=N^{2}\left[-\int \int _{x\neq y}\ln |x-y|\mathrm {d} \nu (x)\mathrm {d} \nu (y)+\int Q(x)\mathrm {d} \nu (x)\right],$

the probability measure is now of the form

$\mathrm {d} \mu _{N}(\mu )={\frac {1}{{\widetilde {Z}}_{N}}}e^{-N^{2}I_{Q}(\nu )}\mathrm {d} \lambda,$

where$I_{Q}(\nu )$is the above functional inside the squared brackets.

Let now

$M_{1}(\mathbb {R} )=\left\{\nu:\nu \geq 0,\ \int _{\mathbb {R} }\mathrm {d} \nu =1\right\}$

be the space of one-dimensional probability measures and consider the minimizer

$E_{Q}=\inf \limits _{\nu \in M_{1}(\mathbb {R} )}-\int \int _{x\neq y}\ln |x-y|\mathrm {d} \nu (x)\mathrm {d} \nu (y)+\int Q(x)\mathrm {d} \nu (x).$

For$E_{Q}$there exists a unique equilibrium measure$\nu _{Q}$through theEuler-Lagrange variational conditionsfor some real constant$l$

$2\int _{\mathbb {R} }\log |x-y|\mathrm {d} \nu (y)-Q(x)=l,\quad x\in J$

$2\int _{\mathbb {R} }\log |x-y|\mathrm {d} \nu (y)-Q(x)\leq l,\quad x\in \mathbb {R} \setminus J$

where$J=\bigcup \limits _{j=1}^{q}[a_{j},b_{j}]$is the support of the measure and define

$q(x)=-\left({\frac {Q'(x)}{2}}\right)^{2}+\int {\frac {Q'(x)-Q'(y)}{x-y}}\mathrm {d} \nu _{Q}(y)$.

The equilibrium measure$\nu _{Q}$has the following Radon–Nikodym density

${\frac {\mathrm {d} \nu _{Q}(x)}{\mathrm {d} x}}={\frac {1}{\pi }}{\sqrt {q(x)}}.$

Mesoscopic regime

The typical statement of the Wigner semicircular law is equivalent to the following statement: For eachfixedinterval$[\lambda _{0}-\Delta \lambda,\lambda _{0}+\Delta \lambda ]$centered at a point$\lambda _{0}$,as$N$,the number of dimensions of the gaussian ensemble increases, the proportion of the eigenvalues falling within the interval converges to$\int _{[\lambda _{0}-\Delta \lambda,\lambda _{0}+\Delta \lambda ]}\rho (t)dt$,where$\rho (t)$is the density of the semicircular distribution.

If$\Delta \lambda$can be allowed to decrease as$N$increases, then we obtain strictly stronger theorems, named "local laws" or "mesoscopic regime".

The mesoscopic regime is intermediate between the local and the global. In themesoscopic regime,one is interested in the limit distribution of eigenvalues in a set that shrinks to zero, but slow enough, such that the number of eigenvalues inside$\to \infty$.

For example, the Ginibre ensemble has a mesoscopic law: For any sequence of shrinking disks with areas$u$inside the unite disk, if the disks have area$A_{n}=O(n^{-1+\epsilon })$,the conditional distribution of the spectrum inside the disks also converges to a uniform distribution. That is, if we cut the shrinking disks along with the spectrum falling inside the disks, then scale the disks up to unit area, we would see the spectra converging to a flat distribution in the disks.

Local regime

In thelocal regime,one is interested in the limit distribution of eigenvalues in a set that shrinks so fast that the number of eigenvalues remains$O(1)$.

Typically this means the study of spacings between eigenvalues, and, more generally, in the joint distribution of eigenvalues in an interval of length of order 1/n.One distinguishes betweenbulk statistics,pertaining to intervals inside the support of the limiting spectral measure, andedge statistics,pertaining to intervals near the boundary of the support.

Bulk statistics

Formally, fix$\lambda _{0}$in theinteriorof thesupportof$N(\lambda )$.Then consider thepoint process $$\Xi (\lambda _{0})=\sum _{j}\delta {\Big (}{\cdot }-n\rho (\lambda _{0})(\lambda _{j}-\lambda _{0}){\Big )}~,$$ where$\lambda _{j}$are the eigenvalues of the random matrix.

The point process$\Xi (\lambda _{0})$captures the statistical properties of eigenvalues in the vicinity of$\lambda _{0}$.For theGaussian ensembles,the limit of$\Xi (\lambda _{0})$is known;thus, for GUE it is adeterminantal point processwith the kernel $$K(x,y)={\frac {\sin \pi (x-y)}{\pi (x-y)}}$$ (thesine kernel).

Theuniversalityprinciple postulates that the limit of$\Xi (\lambda _{0})$as$n\to \infty$should depend only on the symmetry class of the random matrix (and neither on the specific model of random matrices nor on$\lambda _{0}$). Rigorous proofs of universality are known for invariant matrix ensemblesand Wigner matrices.

Edge statistics

Main article:Tracy–Widom distribution

One example of edge statistics is theTracy–Widom distribution.

As another example, consider the Ginibre ensemble. It can be real or complex. The real Ginibre ensemble has i.i.d. standard Gaussian entries${\mathcal {N}}(0,1)$,and the complex Ginibre ensemble has i.i.d. standard complex Gaussian entries${\mathcal {N}}(0,1/2)+i{\mathcal {N}}(0,1/2)$.

Now let$G_{n}$be sampled from the real or complex ensemble, and let$\rho (G_{n})$be the absolute value of its maximal eigenvalue:$$\rho (G_{n}):=\max _{j}|\lambda _{j}|$$We have the following theorem for the edge statistics:

Edge statistics of the Ginibre ensemble—For$G_{n}$and$\rho \left(G_{n}\right)$as above, with probability one, $$\lim _{n\rightarrow \infty }{\frac {1}{\sqrt {n}}}\rho \left(G_{n}\right)=1$$

Moreover, if$\gamma _{n}=\log \left({\frac {n}{2\pi }}\right)-2\log(\log(n))$and $$Y_{n}:={\sqrt {4n\gamma _{n}}}\left({\frac {1}{\sqrt {n}}}\rho \left(G_{n}\right)-1-{\sqrt {\frac {\gamma _{n}}{4n}}}\right),$$ then$Y_{n}$converges in distribution to theGumbel law,i.e., the probability measure on$\mathbb {R}$with cumulative distribution function$F_{\mathrm {Gum} }(x)=e^{-e^{-x}}$.

This theorem refines thecircular law of the Ginibre ensemble.In words, the circular law says that the spectrum of${\frac {1}{\sqrt {n}}}G_{n}$almost surely falls uniformly on the unit disc. and the edge statistics theorem states that the radius of the almost-unit-disk is about$1-{\sqrt {\frac {\gamma _{n}}{4n}}}$,and fluctuates on a scale of${\frac {1}{\sqrt {4n\gamma _{n}}}}$,according to the Gumbel law.

Correlation functions

The joint probability density of the eigenvalues of$n\times n$random Hermitian matrices$M\in \mathbf {H} ^{n\times n}$,with partition functions of the form $$Z_{n}=\int _{M\in \mathbf {H} ^{n\times n}}d\mu _{0}(M)e^{{\text{tr}}(V(M))}$$ where $$V(x):=\sum _{j=1}^{\infty }v_{j}x^{j}$$ and$d\mu _{0}(M)$is the standard Lebesgue measure on the space$\mathbf {H} ^{n\times n}$of Hermitian$n\times n$matrices, is given by $$p_{n,V}(x_{1},\dots,x_{n})={\frac {1}{Z_{n,V}}}\prod _{i<j}(x_{i}-x_{j})^{2}e^{-\sum _{i}V(x_{i})}.$$ The$k$-point correlation functions (ormarginal distributions) are defined as $$R_{n,V}^{(k)}(x_{1},\dots,x_{k})={\frac {n!}{(n-k)!}}\int _{\mathbf {R} }dx_{k+1}\cdots \int _{\mathbb {R} }dx_{n}\,p_{n,V}(x_{1},x_{2},\dots,x_{n}),$$ which are skew symmetric functions of their variables. In particular, the one-point correlation function, ordensity of states,is $$R_{n,V}^{(1)}(x_{1})=n\int _{\mathbb {R} }dx_{2}\cdots \int _{\mathbf {R} }dx_{n}\,p_{n,V}(x_{1},x_{2},\dots,x_{n}).$$ Its integral over a Borel set$B\subset \mathbf {R}$gives the expected number of eigenvalues contained in$B$: $$\int _{B}R_{n,V}^{(1)}(x)dx=\mathbf {E} \left(\#\{{\text{eigenvalues in }}B\}\right).$$

The following result expresses these correlation functions as determinants of the matrices formed from evaluating the appropriate integral kernel at the pairs$(x_{i},x_{j})$of points appearing within the correlator.

Theorem[Dyson-Mehta] For any$k$,$1\leq k\leq n$the$k$-point correlation function$R_{n,V}^{(k)}$can be written as a determinant $$R_{n,V}^{(k)}(x_{1},x_{2},\dots,x_{k})=\det _{1\leq i,j\leq k}\left(K_{n,V}(x_{i},x_{j})\right),$$ where$K_{n,V}(x,y)$is the$n$th Christoffel-Darboux kernel $$K_{n,V}(x,y):=\sum _{k=0}^{n-1}\psi _{k}(x)\psi _{k}(y),$$ associated to$V$,written in terms of the quasipolynomials $$\psi _{k}(x)={1 \over {\sqrt {h_{k}}}}\,p_{k}(z)\,e^{-V(z)/2},$$ where$\{p_{k}(x)\}_{k\in \mathbf {N} }$is a complete sequence of monic polynomials, of the degrees indicated, satisfying the orthogonilty conditions $$\int _{\mathbf {R} }\psi _{j}(x)\psi _{k}(x)dx=\delta _{jk}.$$

Other classes of random matrices

Wishart matrices

Main article:Wishart distribution

Wishart matricesaren×nrandom matrices of the formH=XX^*,whereXis ann×mrandom matrix (m≥n) with independent entries, andX^*is itsconjugate transpose.In the important special case considered by Wishart, the entries ofXare identically distributed Gaussian random variables (either real or complex).

Thelimit of the empirical spectral measure of Wishart matriceswas foundbyVladimir MarchenkoandLeonid Pastur.

Random unitary matrices

Main article:Circular ensembles

Non-Hermitian random matrices

Main article:Circular law

Selected bibliography

Books

• Mehta, M.L. (2004).Random Matrices.Amsterdam: Elsevier/Academic Press.ISBN0-12-088409-7.
• Anderson, G.W.; Guionnet, A.; Zeitouni, O. (2010).An introduction to random matrices.Cambridge: Cambridge University Press.ISBN978-0-521-19452-5.
• Akemann, G.; Baik, J.; Di Francesco, P. (2011).The Oxford Handbook of Random Matrix Theory.Oxford: Oxford University Press.ISBN978-0-19-957400-1.
• 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.ISBN978-1-108-76890-0.

Survey articles

• Edelman, A.; Rao, N.R (2005). "Random matrix theory".Acta Numerica.14:233–297.Bibcode:2005AcNum..14..233E.doi:10.1017/S0962492904000236.S2CID16038147.
• Pastur, L.A. (1973). "Spectra of random self-adjoint operators".Russ. Math. Surv.28(1): 1–67.Bibcode:1973RuMaS..28....1P.doi:10.1070/RM1973v028n01ABEH001396.S2CID250796916.
• Diaconis, Persi(2003)."Patterns in eigenvalues: the 70th Josiah Willard Gibbs lecture".Bulletin of the American Mathematical Society.New Series.40(2): 155–178.doi:10.1090/S0273-0979-03-00975-3.MR1962294.
• Diaconis, Persi(2005)."What is... a random matrix?".Notices of the American Mathematical Society.52(11): 1348–1349.ISSN0002-9920.MR2183871.
• Eynard, Bertrand; Kimura, Taro; Ribault, Sylvain (2015-10-15). "Random matrices".arXiv:1510.04430v2[math-ph].

Historic works

• Wigner, E. (1955). "Characteristic vectors of bordered matrices with infinite dimensions".Annals of Mathematics.62(3): 548–564.doi:10.2307/1970079.JSTOR1970079.
• Wishart, J. (1928). "Generalized product moment distribution in samples".Biometrika.20A(1–2): 32–52.doi:10.1093/biomet/20a.1-2.32.
• von Neumann, J.; Goldstine, H.H. (1947)."Numerical inverting of matrices of high order".Bull. Amer. Math. Soc.53(11): 1021–1099.doi:10.1090/S0002-9904-1947-08909-6.

External links

• Fyodorov, Y. (2011)."Random matrix theory".Scholarpedia.6(3): 9886.Bibcode:2011SchpJ...6.9886F.doi:10.4249/scholarpedia.9886.
• Weisstein, E. W."Random Matrix".Wolfram MathWorld.