Large deviations of the shifted index number in the Gaussian ensemble

We show that, using the Coulomb fluid approach, we are able to derive a rate function $\Psi(c,x)$ of two variables that captures: (i) the large deviations of bulk eigenvalues; (ii) the large deviations of extreme eigenvalues (both left and right large deviations); (iii) the statistics of the fraction $c$ of eigenvalues to the left of a position $x$. Thus, $\Psi(c,x)$ explains the full order statistics of the eigenvalues of large random Gaussian matrices as well as the statistics of the shifted index number. All our analytical findings are thoroughly compared with Monte Carlo simulations, obtaining excellent agreement. A summary of preliminary results was already presented in [22] in the context of one-dimensional trapped spinless fermions in a harmonic potential.