Random matrix

In

can be modeled mathematically as problems concerning large, random matrices.

Applications

Physics

In

mean-field approximation

.

In quantum 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.^[3]

In quantum optics, transformations described by random unitary matrices are crucial for demonstrating the advantage of quantum over classical computation (see, e.g., the boson sampling model).^[5] Moreover, such random unitary transformations can be directly implemented in an optical circuit, by mapping their parameters to optical circuit components (that is beam splitters and phase shifters).^[6]

Random matrix theory has also found applications to the chiral Dirac operator in

Mathematical statistics and numerical analysis

In multivariate statistics, random matrices were introduced by John Wishart, who sought to estimate covariance matrices of large samples.^[15] Chernoff-, Bernstein-, and Hoeffding-type inequalities can typically be strengthened when applied to the maximal eigenvalue (i.e. the eigenvalue of largest magnitude) of a finite sum of random Hermitian matrices.^[16] Random matrix theory is used to study the spectral properties of random matrices—such as sample covariance matrices—which is of particular interest in high-dimensional statistics. Random matrix theory also saw applications in neuronal networks^[17] and deep 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.^[18]

In numerical analysis, random matrices have been used since the work of John von Neumann and Herman Goldstine^[19] to describe computation errors in operations such as matrix multiplication. Although random entries are traditional "generic" inputs to an algorithm, the concentration of measure associated with random matrix distributions implies that random matrices will not test large portions of an algorithm's input space.^[20]

Number theory

In

.

Free probability

The relation of free probability with random matrices^[22] is 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;^[23] 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 chaos^[24] when 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 variation^[25][26] and time to synchrony depends on network topology.^[27][28]

In the analysis of massive data such as

).

Optimal control

In

(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

In computational 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,^[33] 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.^{[citation needed]}

History

This section needs expansion. You can help by adding to it. (April 2024)

Random matrix theory first gained attention beyond mathematics literature in the context of nuclear physics. Experiments by

Gaussian ensembles

The most-commonly studied random matrix distributions are the Gaussian ensembles: GOE, GUE and GSE. 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.

Definitions

The Gaussian unitary ensemble ${\displaystyle {\text{GUE}}(n)}$ is described by the Gaussian measure with density

on the space of ${\displaystyle n\times n}$ ${\displaystyle H=(H_{ij})_{i,j=1}^{n}}$. Here is a normalization constant, chosen so that the integral of the density is equal to one. The term unitary refers to the fact that the distribution is invariant under unitary conjugation. The Gaussian unitary ensemble models Hamiltonians lacking time-reversal symmetry.

The Gaussian orthogonal ensemble ${\displaystyle {\text{GOE}}(n)}$ is described by the Gaussian measure with density

on the space of n × n real symmetric matrices H = (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 ${\displaystyle H=(G+G^{T})/{\sqrt {2n}}}$, where ${\displaystyle G}$ is an ${\displaystyle n\times n}$ matrix with IID samples from the standard normal distribution.

The Gaussian symplectic ensemble ${\displaystyle {\text{GSE}}(n)}$ is described by the Gaussian measure with density

on the space of n × n Hermitian quaternionic matrices, e.g. symmetric square matrices composed of quaternions, H = (H_ij)ⁿ
_i,j=1. Its distribution is invariant under conjugation by the symplectic 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

from which all higher correlations follow by Isserlis' theorem.

Moment generating functions

The moment generating function for the GOE is

where ${\displaystyle \|\cdot \|_{F}}$ is the Frobenius norm.

Spectral density

The joint

λ₁, λ₂, ..., λ_n of GUE/GOE/GSE is given by

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

(1)

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

The distribution of the largest eigenvalue for GOE, and GUE, are explicitly solvable.^[35] They converge to the Tracy–Widom distribution after shifting and scaling appropriately.

Convergence to Wigner semicircular distribution

The spectrum, divided by ${\displaystyle {\sqrt {N\sigma ^{2}}}}$, converges in distribution to the semicircular distribution on the interval ${\displaystyle [-2,+2]}$: ${\displaystyle \rho (x)={\frac {1}{2\pi }}{\sqrt {4-x^{2}}}}$. Here ${\displaystyle \sigma ^{2}}$ 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 ${\displaystyle \lambda _{1}<\ldots <\lambda _{n}<\lambda _{n+1}<\ldots }$, one defines the normalized spacings ${\displaystyle s=(\lambda _{n+1}-\lambda _{n})/\langle s\rangle }$, where ${\displaystyle \langle s\rangle =\langle \lambda _{n+1}-\lambda _{n}\rangle }$ is the mean spacing. The probability distribution of spacings is approximately given by,

for the orthogonal ensemble GOE ${\displaystyle \beta =1}$, for the unitary ensemble GUE ${\displaystyle \beta =2}$, and for the symplectic ensemble GSE ${\displaystyle \beta =4}$.

The numerical constants are such that ${\displaystyle p_{\beta }(s)}$ is normalized:

and the mean spacing is, for ${\displaystyle \beta =1,2,4}$.

Generalizations

Wigner matrices are random Hermitian matrices ${\textstyle H_{n}=(H_{n}(i,j))_{i,j=1}^{n}}$ such that the entries

above the main diagonal are independent random variables with zero mean and have identical second moments.

Invariant matrix ensembles are 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 function V is 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.^[36][37]

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. ^[38]

Empirical spectral measure

The empirical spectral measure μ_H of H is defined by

Usually, the limit of ${\displaystyle \mu _{H}}$ is a deterministic measure; this is a particular case of self-averaging. The cumulative distribution function of the limiting measure is called the integrated density of states and is denoted N(λ). If the integrated density of states is differentiable, its derivative is called the density of states and is denoted ρ(λ).

Alternative expressions

Types of convergence

Given a matrix ensemble, we say that its spectral measures converge weakly to ${\displaystyle \rho }$ iff for any measurable set ${\displaystyle A}$, the ensemble-average converges:

Convergence weakly almost surely: If we sample ${\displaystyle H_{1},H_{2},H_{3},\dots }$ independently from the ensemble, then with probability 1,for any measurable set ${\displaystyle A}$.

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

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 ${\displaystyle \{G_{i,j}\}_{i,j=1,2,3,\dots }}$.
• Define each ${\displaystyle H_{n}=(G_{n}+G_{n}^{T})/{\sqrt {2n}}}$ where ${\displaystyle G_{n}}$ is the matrix made of entries ${\displaystyle \{G_{i,j}\}_{i,j=1,2,\dots ,n}}$.

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

The limit of the empirical spectral measure for Wigner matrices was described by Eugene Wigner; see Wigner semicircle distribution and Wigner surmise. As far as sample covariance matrices are concerned, a theory was developed by Marčenko and Pastur.^[39][40]

The limit of the empirical spectral measure of invariant matrix ensembles is described by a certain integral equation which arises from potential theory.^[41]

Fluctuations

For the linear statistics N_f,H = n⁻¹ Σ f(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

is known.^[42][43]

The variational problem for the unitary ensembles

Consider the measure

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

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

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

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

the probability measure is now of the form

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

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

Let now

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

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

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

For ${\displaystyle E_{Q}}$ there exists a unique equilibrium measure ${\displaystyle \nu _{Q}}$ through the Euler-Lagrange variational conditions for some real constant ${\displaystyle l}$

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

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

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

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

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

${\displaystyle {\frac {\mathrm {d} \nu _{Q}(x)}{\mathrm {d} x}}={\frac {1}{\pi }}{\sqrt {q(x)}}.}$^[44]

Mesoscopic regime

^[45][46] The typical statement of the Wigner semicircular law is equivalent to the following statement: For each fixed interval ${\displaystyle [\lambda _{0}-\Delta \lambda ,\lambda _{0}+\Delta \lambda ]}$ centered at a point ${\displaystyle \lambda _{0}}$, as ${\displaystyle N}$, the number of dimensions of the gaussian ensemble increases, the proportion of the eigenvalues falling within the interval converges to ${\displaystyle \int _{[\lambda _{0}-\Delta \lambda ,\lambda _{0}+\Delta \lambda ]}\rho (t)dt}$, where ${\displaystyle \rho (t)}$ is the density of the semicircular distribution.

If ${\displaystyle \Delta \lambda }$ can be allowed to decrease as ${\displaystyle N}$ 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 the mesoscopic 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 ${\displaystyle \to \infty }$.

For example, the Ginibre ensemble has a mesoscopic law: For any sequence of shrinking disks with areas ${\displaystyle u}$inside the unite disk, if the disks have area ${\displaystyle A_{n}=O(n^{-1+\epsilon })}$, 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.^[46]

Local regime

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

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 between bulk statistics, pertaining to intervals inside the support of the limiting spectral measure, and edge statistics, pertaining to intervals near the boundary of the support.

Bulk statistics

Formally, fix ${\displaystyle \lambda _{0}}$ in the interior of the support of ${\displaystyle N(\lambda )}$. Then consider the point process

where ${\displaystyle \lambda _{j}}$ are the eigenvalues of the random matrix.

The point process ${\displaystyle \Xi (\lambda _{0})}$ captures the statistical properties of eigenvalues in the vicinity of ${\displaystyle \lambda _{0}}$. For the Gaussian ensembles, the limit of ${\displaystyle \Xi (\lambda _{0})}$ is known;^[4] thus, for GUE it is a determinantal point process with the kernel

(the sine kernel).

The universality principle postulates that the limit of ${\displaystyle \Xi (\lambda _{0})}$ as ${\displaystyle n\to \infty }$ should depend only on the symmetry class of the random matrix (and neither on the specific model of random matrices nor on ${\displaystyle \lambda _{0}}$). Rigorous proofs of universality are known for invariant matrix ensembles^[47][48] and Wigner matrices.^[49][50]

Edge statistics

One example of edge statistics is the Tracy–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 ${\displaystyle {\mathcal {N}}(0,1)}$, and the complex Ginibre ensemble has i.i.d. standard complex Gaussian entries ${\displaystyle {\mathcal {N}}(0,1/2)+i{\mathcal {N}}(0,1/2)}$.

Now let ${\displaystyle G_{n}}$ be sampled from the real or complex ensemble, and let ${\displaystyle \rho (G_{n})}$ be the absolute value of its maximal eigenvalue:

We have the following theorem for the edge statistics:^[51]

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

Correlation functions

The joint probability density of the eigenvalues of ${\displaystyle n\times n}$ random Hermitian matrices ${\displaystyle M\in \mathbf {H} ^{n\times n}}$, with partition functions of the form

where and ${\displaystyle d\mu _{0}(M)}$ is the standard Lebesgue measure on the space ${\displaystyle \mathbf {H} ^{n\times n}}$ of Hermitian ${\displaystyle n\times n}$ matrices, is given by The ${\displaystyle k}$-point correlation functions (or marginal distributions) are defined as which are skew symmetric functions of their variables. In particular, the one-point correlation function, or density of states, is Its integral over a Borel set ${\displaystyle B\subset \mathbf {R} }$ gives the expected number of eigenvalues contained in ${\displaystyle B}$:

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

Theorem [Dyson-Mehta] For any ${\displaystyle k}$, ${\displaystyle 1\leq k\leq n}$ the ${\displaystyle k}$-point correlation function ${\displaystyle R_{n,V}^{(k)}}$ can be written as a determinant

where ${\displaystyle K_{n,V}(x,y)}$ is the ${\displaystyle n}$th Christoffel-Darboux kernel associated to ${\displaystyle V}$, written in terms of the quasipolynomials where ${\displaystyle \{p_{k}(x)\}_{k\in \mathbf {N} }}$ is a complete sequence of monic polynomials, of the degrees indicated, satisfying the orthogonilty conditions

Other classes of random matrices

Wishart matrices

Wishart matrices are n × n random matrices of the form H = X X^*, where X is an n × m random matrix (m ≥ n) with independent entries, and X^* is its conjugate transpose. In the important special case considered by Wishart, the entries of X are identically distributed Gaussian random variables (either real or complex).

The

.

Random unitary matrices

Main article:

Circular ensembles

Non-Hermitian random matrices

Selected bibliography

Books

• Mehta, M.L. (2004). Random Matrices. Amsterdam: Elsevier/Academic Press.
  ISBN 0-12-088409-7
  .
• Anderson, G.W.; Guionnet, A.; Zeitouni, O. (2010). An introduction to random matrices. Cambridge: Cambridge University Press.
  ISBN 978-0-521-19452-5
  .
• Akemann, G.; Baik, J.; Di Francesco, P. (2011). The Oxford Handbook of Random Matrix Theory. Oxford: Oxford University Press.
  ISBN 978-0-19-957400-1
  .
• Potters, Marc; Bouchaud, Jean-Philippe (2020-11-30). A First Course in Random Matrix Theory: for Physicists, Engineers and Data Scientists. Cambridge University Press.
  ISBN 978-1-108-76890-0
  .

Survey articles

• Edelman, A.; Rao, N.R (2005). "Random matrix theory". Acta Numerica. 14: 233–297.
  S2CID 16038147
  .
• Pastur, L.A. (1973). "Spectra of random self-adjoint operators". Russ. Math. Surv. 28 (1): 1–67.
  S2CID 250796916
  .
• MR 1962294
  .
• MR 2183871
  .
• 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.
  JSTOR 1970079
  .
• 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
  .

References

External links

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