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Instantons and Extreme Value Statistics of Random Matrices

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Atkin, Max [UCL]

We discuss the distribution of the largest eigenvalue of a random N x N Hermitian matrix. Utilising results from the quantum gravity and string theory literature it is seen that the orthogonal polynomials approach, first introduced by Majumdar and Nadal, can be extended to calculate both the left and right tail large deviations of the maximum eigenvalue. This framework does not only provide computational advantages when considering the left and right tail large deviations for general potentials, as is done explicitly for the first multi-critical potential, but it also offers an interesting interpretation of the results. In particular, it is seen that the left tail large deviations follow from a standard perturbative large N expansion of the free energy, while the right tail large deviations are related to the non-perturbative expansion and thus to instanton corrections. Considering the standard interpretation of instantons as tunnelling of eigenvalues, we see that the right tail rate function can be identified with the instanton action which in turn can be given as a simple expression in terms of the spectral curve. From the string theory point of view these non-perturbative corrections correspond to branes and can be identified with FZZT branes.

Document type Article de périodique (Journal article) – Article de recherche
Access type Accès libre
Publication date 2014
Language Anglais
Journal information "Journal of High Energy Physics" - Vol. 2014, no. 04, p. 118 (2014)
Peer reviewed yes
e-issn 1029-8479
Publication status Publié
Affiliation UCL - SST/IRMP - Institut de recherche en mathématique et physique
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Bibliographic reference Atkin, Max. Instantons and Extreme Value Statistics of Random Matrices. In: Journal of High Energy Physics, Vol. 2014, no. 04, p. 118 (2014)
Permanent URL http://hdl.handle.net/2078.1/156730