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,[1] or the Hermite ensemble.[2]
Definitions
[edit]Conventions
[edit]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.
- : a positive real number. Called the Dyson index. The cases of are special.
- : the side-length of a matrix. Always a positive integer.
- : a matrix sampled from a Gaussian ensemble with size . The letter stands for "Wigner".
- : the adjoint of a matrix. We assume (self-adjoint) when is sampled from a gaussian ensemble.
- If is real, then is its transpose.
- If is complex or quaternionic, then is its conjugate transpose.
- : the eigenvalues of the matrix, which are all real, since the matrices are always assumed to be self-adjoint.
- : the variance of on-diagonal matrix entries. We assume that for each , all on-diagonal matrix entries have the same variance. It is always defined as .
- : the variance of off-diagonal matrix entries. We assume that for each , all off-diagonal matrix entries have the same variance. It is always defined as where .
- For a complex number, .
- For a quaternion, .
- : the partition function.
Name | GOE(N) | GUE(N) | GSE(N) | GβE(N) |
---|---|---|---|---|
Full name | Gaussian orthogonal ensemble | Gaussian unitary ensemble | Gaussian symplectic ensemble | Gaussian beta ensemble |
1 | 2 | 4 | β | |
2 | 1 | 1/2 | 2/β | |
1 | 1 | 1 | 1 | |
matrix density | ||||
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.
Name | ||
---|---|---|
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) |
There are equivalent definitions for the GβE(N) ensembles, given below.
By sampling
[edit]For all cases, the GβE(N) ensemble is defined by how it is sampled:
- Sample a gaussian matrix , such that all its entries are IID sampled from the corresponding standard normal distribution.
- If , then .
- If , then .
- If , then .
- Let .
By density
[edit]For all cases, the GβE(N) ensemble is defined with density function where the partition function is .
The Gaussian orthogonal ensemble GOE(N) is defined as the probability distribution over symmetric matrices with density functionwhere the partition function is .
Explicitly, since there are only degrees of freedom, the parameterization is as follows:where we pick the upper diagonal entries as the degrees of freedom.
The Gaussian unitary ensemble GUE(N) is defined as the probability distribution over Hermitian matrices with density functionwhere the partition function is .
Explicitly, since there are only degrees of freedom, the parameterization is as follows: where we pick the upper diagonal entries as the degrees of freedom.
The Gaussian symplectic ensemble GSE(N) is defined as the probability distribution over self‑adjoint quaternionic matrices with density functionwhere the partition function is .
Explicitly, since there are only degrees of freedom, the parameterization is as follows:where we write and pick the upper diagonal entries as the degrees of freedom.
By invariance
[edit]For all cases, the GβE(N) ensemble is uniquely characterized (up to affine transform) by its symmetries, or invariance under appropriate transformations.[3]
For GOE, consider a probability distribution over symmetric matrices satisfying the following properties:
- Invariance under orthogonal transformation: For any fixed (not random) orthogonal matrix , let be a random sample from the distribution. Then has the same distribution as .
- Independence: The entries are independently distributed.
For GUE, consider a probability distribution over Hermitian matrices satisfying the following properties:
- Invariance under unitary transformation: For any fixed (not random) unitary matrix , let be a random sample from the distribution. Then has the same distribution as .
- Independence: The entries are independently distributed.
For GSE, consider a probability distribution over self-adjoint quaternionic matrices satisfying the following properties:
- Invariance under symplectic transformation: For any fixed (not random) symplectic matrix , let be a random sample from the distribution. Then has the same distribution as .
- Independence: The entries are independently distributed.
In all 3 cases, these conditions force the distribution to have the form , where and . Thus, with the further specification of , we recover the GOE, GUE, GSE.[4]
More succinctly stated, each of GOE, GUE, GSE is uniquely specified by invariance, independence, the mean, and the variance.
By spectral distribution
[edit]For all cases, the GβE(N) ensemble is defined as the ensemble obtained by , where
- is a diagonal real matrix with its entries sampled according to the spectral density, defined below;
- is an orthogonal/unitary/symplectic matrix sampled uniformly, that is, from the normalized Haar measure of the orthogonal/unitary/symplectic group.
Spectral density
[edit]For eigenvalues the joint density of GβE(N) iswhere is the Vandermonde determinant, and the partition function is explicitly evaluated as a Selberg integral:[5]where is the Euler Gamma function.
Determinantal point process
[edit]
Define functions , where 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 .
This allows a closed-form solution of the spectral density of GUE(N) for finite values of :[6]The spectral distribution of can also be written as a quaternionic determinantal point process involving skew-orthogonal polynomials.[7][8]
Tridiagonalization
[edit]For all cases, given a sampled matrix from the GβE(N) ensemble, we can perform a Householder transformation tridiagonalization on it to obtain a tridiagonal matrix , which has the same distribution aswhere each is gaussian-distributed, and each is chi-distributed, and all are independent.
In particular, this definition allows extension to all cases, leading to the gaussian beta ensembles.[9][10]
Global law
[edit]
The Wigner semicircle law states that the empirical eigenvalue distribution of converges in distribution to the Wigner semicircle distribution with radius 2.[11][12] That is, the distribution on with probability density function
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
[edit]The joint density can be written as a Gibbs measure:with the energy function (also called the Hamiltonian) . This can be interpreted physically as a Boltzmann distribution of a physical system consisting of identical unit electric charges constrained to move on the real line, repelling each other via the two-dimensional Coulomb potential , while being attracted to the origin via a quadratic potential . This is the Coulomb gas model for the eigenvalues.
In the macroscopic limit, one rescales and defines the empirical measure , obtaining , where the mean-field functional .
The minimizer of over probability measures is the Wigner semicircle law , which gives the limiting eigenvalue density.[13] The value yields the leading order term in , termed the Coulomb gas free energy.
Alternatively, suppose that there exists a , such that the quadratic electric potential can be recreated (up to an additive constant) viaThen, imposing a fixed background negative electric charge of density exactly cancels out the electric repulsion between the freely moving positive charges. Such a function does exist: , which can be found by solving an integral equation. This indicates that the Wigner semicircle distribution is the equilibrium distribution.[14][15][16]
Gaussian fluctuations about obtained by expanding to second order produce the sine kernel in the bulk and the Airy kernel at the soft edge after proper rescaling.
Mesoscopic law
[edit]Microscopic law
[edit]Extreme value
[edit]The largest eigenvalue for GOE and GUE follows the Tracy–Widom distribution after proper centering and scaling.[17]
Level spacings
[edit]From ordered eigenvalues , define normalized spacings with mean spacing . This normalizes the spacings by:With this, the approximate spacing distributions are
Other properties
[edit]Point correlation
[edit]Matrix elements satisfy and with higher correlations given by Isserlis' theorem.
Moment generating functions
[edit]For GOE, , where is the Frobenius norm.
See also
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References
[edit]- ^ (Potters & Bouchaud 2020)
- ^ Edelman, Alan; Sutton, Brian D. (June 2007). "From Random Matrices to Stochastic Operators". Journal of Statistical Physics. 127 (6): 1121–1165. doi:10.1007/s10955-006-9226-4. ISSN 0022-4715.
- ^ (Mehta 2004, Def 2.3.1, 2.4.1, 2.5.1)
- ^ (Mehta 2004, Thm. 2.6.3)
- ^ (Mehta 2004, Thm. 3.3.1)
- ^ (Livan, Novaes & Vivo 2018, Chap. 10)
- ^ (Livan, Novaes & Vivo 2018, Chap. 12)
- ^ Adler, M.; Forrester, P. J.; Nagao, T.; van Moerbeke, P. (2000-04-01). "Classical Skew Orthogonal Polynomials and Random Matrices". Journal of Statistical Physics. 99 (1): 141–170. doi:10.1023/A:1018644606835. ISSN 1572-9613.
- ^ (Forrester 2010, Prop. 1.9.4)
- ^ Dumitriu, Ioana; Edelman, Alan (2002-11-01). "Matrix models for beta ensembles". Journal of Mathematical Physics. 43 (11): 5830–5847. doi:10.1063/1.1507823. ISSN 0022-2488.
- ^ (Tao 2012, Thm. 2.4.2)
- ^ (Forrester 2010, Prop. 1.4.4)
- ^ (Tao 2012, Sec. 2.6.3.)
- ^ (Livan, Novaes & Vivo 2018, Sec. 5.2)
- ^ Forrester, Peter (2015-09-17), Akemann, Gernot; Baik, Jinho; Di Francesco, Philippe (eds.), "Beta ensembles", The Oxford Handbook of Random Matrix Theory, Oxford University Press, p. 0, ISBN 978-0-19-874419-1
- ^ (Forrester 2010, Sec.1.4)
- ^ Chiani M (2014). "Distribution of the largest eigenvalue for real Wishart and Gaussian random matrices and a simple approximation for the Tracy–Widom distribution". Journal of Multivariate Analysis. 129: 69–81. arXiv:1209.3394. doi:10.1016/j.jmva.2014.04.002. S2CID 15889291.
- Deift, Percy (2000). Orthogonal polynomials and random matrices: a Riemann-Hilbert approach. Courant lecture notes in mathematics. Providence, R.I: American Mathematical Society. ISBN 978-0-8218-2695-9.
- Mehta, M.L. (2004). Random Matrices. Amsterdam: Elsevier/Academic Press. ISBN 0-12-088409-7.
- Deift, Percy; Gioev, Dimitri (2009). Random matrix theory: invariant ensembles and universality. Courant lecture notes in mathematics. New York : Providence, R.I: Courant Institute of Mathematical Sciences ; American Mathematical Society. ISBN 978-0-8218-4737-4.
- Forrester, Peter (2010). Log-gases and random matrices. London Mathematical Society monographs. Princeton: Princeton University Press. ISBN 978-0-691-12829-0.
- Anderson, G.W.; Guionnet, A.; Zeitouni, O. (2010). An introduction to random matrices. Cambridge: Cambridge University Press. ISBN 978-0-521-19452-5.
- Bai, Zhidong; Silverstein, Jack W. (2010). Spectral analysis of large dimensional random matrices. Springer series in statistics (2 ed.). New York ; London: Springer. doi:10.1007/978-1-4419-0661-8. ISBN 978-1-4419-0660-1. ISSN 0172-7397.
- 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.
- Tao, Terence (2012). Topics in random matrix theory. Graduate studies in mathematics. Providence, R.I: American Mathematical Society. ISBN 978-0-8218-7430-1.
- Mingo, James A.; Speicher, Roland (2017). Free Probability and Random Matrices. Fields Institute Monographs. New York, NY: Springer. ISBN 978-1-4939-6942-5.
- Livan, Giacomo; Novaes, Marcel; Vivo, Pierpaolo (2018). Introduction to Random Matrices. SpringerBriefs in Mathematical Physics. Vol. 26. Cham: Springer International Publishing. 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.