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
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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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