Abstract
Let us continue the study of the Coulomb gas method for large random matrices.
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- 1.
The pre-factor \(C_{N,\beta }\) has the large-N behavior (4.26), whose logarithm is \(\sim \mathcal {O}(N^2\ln N)\) and thus strictly speaking leading with respect to \(N^2\). However, it is just an overall constant term, and the ‘dynamical’ part of the free energy is of \(\sim \mathcal {O}(N^2)\).
- 2.
Note that the factor 1 / 2 in front of the double integral disappears because the functional differentiation picks up two counting functions, as in the integrand we have \(n(x)n(x^\prime )\). An interesting account on functional differentiation can be found at [1].
- 3.
This is true in general for potentials growing super-logarithmically at infinity—not just for the quadratic potential corresponding to Gaussian ensembles.
- 4.
This means precisely the limit \(\lim _{\varepsilon \rightarrow 0}\left[ \int ^{x-\varepsilon }F(x^\prime )dx^\prime + \int _{x+\varepsilon }F(x^\prime )dx^\prime \right] \), if x is a singular point of F(x).
- 5.
It may be useful to first change variables \(z=(x-a)/(b-a)\). The resulting integrals can then be handled by most symbolic computation programs.
- 6.
The fact that the soft edges are symmetrically located around the origin is a consequence of the symmetry of the confining potential under the exchange \(x\rightarrow -x\).
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Livan, G., Novaes, M., Vivo, P. (2018). Saddle-Point-of-View. In: Introduction to Random Matrices. SpringerBriefs in Mathematical Physics, vol 26. Springer, Cham. https://doi.org/10.1007/978-3-319-70885-0_5
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DOI: https://doi.org/10.1007/978-3-319-70885-0_5
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