Abstract
In this chapter, we start discussing the eigenvalues of random matrices.
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Notes
- 1.
Why is it defined a ‘surmise’? After all, it is the result of an exact calculation! The story goes as follows: at a conference on Neutron Physics by Time-of-Flight, held at the Oak Ridge National Laboratory in 1956, people asked a question about the possible shape of the distribution of the spacings of energy levels in a heavy nucleus. E. P. Wigner, who was in the audience, walked up to the blackboard and guessed (=surmised) the answer given above.
- 2.
We will use the same symbol \(\rho \) for both the jpdf of the entries in the upper triangle and of the eigenvalues.
- 3.
- 4.
It can be computed via the so-called Mehta’s integral, a close relative of the celebrated Selberg’s integral [4].
- 5.
The Dyson index is equal to the number of real variables needed to specify one entry of your matrix: 1 for real, 2 for complex and 4 for quaternions. This is usually referred to as Dyson’s threefold way. For the Gaussian ensemble, then, GOE corresponds to \(\beta =1\), GUE to \(\beta =2\) and GSE to \(\beta =4\).
- 6.
For \(\beta =4\), each matrix has 2N eigenvalues that are two-fold degenerate.
- 7.
Quite often, however, you find in the literature a Gaussian weight including extra factors, such as \(\exp (-(\beta /2)\sum _ix_i^2)\) or \(\exp (-(N/2)\sum _ix_i^2)\). One then needs to be very careful when comparing theoretical results (obtained with such conventions) to numerical simulations—in particular, a rescaling of the numerical eigenvalues by \(\sqrt{\beta }\) or \(\sqrt{N}\) before histogramming is essential in these two modified scenarios.
References
P. Šeba, J. Stat. Mech. L10002 (2009)
P.L. Hsu, Ann. Hum. Genet. 9, 250 (1939)
C.E. Porter, N. Rosenzweig, Ann. Acad. Sci. Fennicae, Serie A VI Physica 44 (1960), reprinted in C.E. Porter, Statistical Theories of Spectra: Fluctuations (Academic Press, New York, 1965)
P.J. Forrester, S.O. Warnaar, Bull. Amer. Math. Soc. (N.S) 45, 489 (2008)
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Livan, G., Novaes, M., Vivo, P. (2018). Value the Eigenvalue. In: Introduction to Random Matrices. SpringerBriefs in Mathematical Physics, vol 26. Springer, Cham. https://doi.org/10.1007/978-3-319-70885-0_2
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