  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.
