Abstract
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.
Highlights
Behaviour of the eigenvalues as N → ∞ is characterised by them being confined to a finite number of finite length intervals on the real axis with a density given by the spectral density function ρ(x)
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 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 offers an interesting interpretation of the results
Summary
First we recall from the standard matrix model literature (see e.g. [1, 23, 24] for reviews) that we may integrate out the angular degrees of freedom in the matrix M to obtain an expression for the partition function related to the Gibbs measure (1.1) only in terms of. Note that the coefficients in the expression for each πn depend on z, t and the potential It is a standard result of random matrix theory that the Vandermonde determinant may be written in terms of these polynomials, thereby allowing us to evaluate the partition function. In [9] recursion relations for rn were obtained in the case of a Gaussian potential. They had the unpleasant property of containing derivatives of z which makes them difficult to use for potentials of arbitrary order. One of the advances made in [6] was to obtain a set of purely algebraic recursion relations for any potential, which is what we will refer to as the string equations.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
