# Instantons and Extreme Value Statistics of Random Matrices

Atkin MR, Zohren S (2014)

Journal of High Energy Physics (4).

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Author

Atkin, Max R.
;
Zohren, Stefan

Abstract

We discuss the distribution of the largest eigenvalue of a random N x NHermitian matrix. Utilising results from the quantum gravity and string theoryliterature it is seen that the orthogonal polynomials approach, firstintroduced by Majumdar and Nadal, can be extended to calculate both the leftand right tail large deviations of the maximum eigenvalue. This framework doesnot only provide computational advantages when considering the left and righttail large deviations for general potentials, as is done explicitly for thefirst multi-critical potential, but it also offers an interestinginterpretation of the results. In particular, it is seen that the left taillarge deviations follow from a standard perturbative large N expansion of thefree energy, while the right tail large deviations are related to thenon-perturbative expansion and thus to instanton corrections. Considering thestandard interpretation of instantons as tunnelling of eigenvalues, we see thatthe right tail rate function can be identified with the instanton action whichin turn can be given as a simple expression in terms of the spectral curve.From the string theory point of view these non-perturbative correctionscorrespond to branes and can be identified with FZZT branes.

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Financial disclosure

Open Access publishing charge funded by SCOAP$^3$ – Sponsoring Consortium for Open Access Publishing in Particle Physics and Bielefeld University.

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### Cite this

Atkin MR, Zohren S. Instantons and Extreme Value Statistics of Random Matrices.

*Journal of High Energy Physics*. 2014;(4).Atkin, M. R., & Zohren, S. (2014). Instantons and Extreme Value Statistics of Random Matrices.

*Journal of High Energy Physics*(4).Atkin, M. R., and Zohren, S. (2014). Instantons and Extreme Value Statistics of Random Matrices.

*Journal of High Energy Physics*.Atkin, M.R., & Zohren, S., 2014. Instantons and Extreme Value Statistics of Random Matrices.

*Journal of High Energy Physics*, (4).M.R. Atkin and S. Zohren, “Instantons and Extreme Value Statistics of Random Matrices”,

*Journal of High Energy Physics*, 2014.Atkin, M.R., Zohren, S.: Instantons and Extreme Value Statistics of Random Matrices. Journal of High Energy Physics. (2014).

Atkin, Max R., and Zohren, Stefan. “Instantons and Extreme Value Statistics of Random Matrices”.

*Journal of High Energy Physics*4 (2014).
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arXiv 1307.3118

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