Abstract
In this Chapter, we continue setting up the formalism and provide a simple classification of matrix models.
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Notes
- 1.
As we know, the Dirac delta function (or rather distribution) \(\delta (x)\) is basically an extremely peaked function at the point \(x=0\), like the limit of a Gaussian pdf as its variance goes to zero, \(\delta (x)=\lim _{\varepsilon \rightarrow 0^+} \frac{1}{2\sqrt{\pi \varepsilon }}e^{-x^2/(4\varepsilon )}\).
- 2.
Compute
$$\begin{aligned} N\int _a^b n(x)dx= \sum _{i=1}^N\int _a^b \delta (x-x_i)dx=\sum _{i=1}^N \chi _{[a,b]}(x_i)\ , \end{aligned}$$(3.5)where the indicator function \(\chi _{[a,b]}(z)\) is equal to 1 if \(z\in (a,b)\) and 0 otherwise. This is by definition the number of eigenvalues between a and b, as it should.
- 3.
We use again the shorthand \(d\varvec{x}=\prod _{j=1}^N dx_j\).
- 4.
U is orthogonal/unitary/symplectic if H is real symmetric/complex hermitian/quaternion self-dual, respectively. You surely have noticed that this is precisely the origin of the names given to the ensembles: Orthogonal, Unitary and Symplectic.
- 5.
In its three incarnations: GOE, GUE and GSE.
References
R. Kühn, J. Phys. A: Math. Theor. 41, 295002 (2008)
P. Cizeau, J.P. Bouchaud, Phys. Rev. E 50, 1810 (1994)
A.D. Mirlin, Y.V. Fyodorov, F.-M. Dittes, J. Quezada, T.H. Seligman, Phys. Rev. E 54, 3221 (1996)
H. Weyl, Classical Groups (Princeton Univ. Press, Princeton, 1946)
K.A. Muttalib, Y. Chen, M.E.H. Ismail, V.N. Nicopoulos, Phys. Rev. Lett. 71, 471 (1993)
C.E. Porter and N. Rosenzweig, Annals of the Acad. Sci. Fennicae, Serie A VI Physica 44, 1 (1960), reprinted in C.E. Porter, Statistical Theories of Spectra: Fluctuations (Academic Press, New York, 1965)
A. Borodin, Nuclear Physics B 536, 704 (1998)
P. Desrosiers, P.J. Forrester, J. Approx. Theory 152, 167 (2008)
J.T. Albrecht, C.P. Chan, A. Edelman, Found. Comput. Math. 9, 461 (2008)
Y.V. Fyodorov, Introduction to the Random Matrix Theory: Gaussian Unitary Ensemble and Beyond (2004), https://arxiv.org/pdf/math-ph/0412017.pdf
M. Zirnbauer, Symmetry classes in random matrix theory (2004), https://arxiv.org/pdf/math-ph/0404058.pdf
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Livan, G., Novaes, M., Vivo, P. (2018). Classified Material. In: Introduction to Random Matrices. SpringerBriefs in Mathematical Physics, vol 26. Springer, Cham. https://doi.org/10.1007/978-3-319-70885-0_3
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