A unified approach for large deviations of bulk and extreme eigenvalues of the Wishart ensemble

Within the framework of the Coulomb fluid picture, we present a unified approach to derive the large deviations of bulk and extreme eigenvalues of large Wishart matrices. By analysing the statistics of the shifted index number we are able to derive a rate function $Ψ(c, x)$ depending on two variables: the fraction $c$ of eigenvalues to the left of an infinite energetic barrier at position $x$. For a fixed value of $c$, the rate function gives the large deviations of the bulk eigenvalues. In particular, in the limits $c\to 0$ or $c\to 1$ it is possible to extract the left and right deviations of the smallest and largest eigenvalues, respectively. Alternatively, for a fixed value $x$ of the barrier, the rate function provides the large deviations of the shifted index number. All our analytical findings are compared with Metropolis Monte Carlo simulations, obtaining excellent agreement.

No reviews yet.

No comments yet.