Spectral order statistics of Gaussian random matrices: large deviations for trapped fermions and associated phase transitions

We compute the full order statistics of a one-dimensional gas of fermions in a harmonic trap at zero temperature, including its large deviation tails. The problem amounts to computing the probability distribution of the $k$th smallest eigenvalue $\lambda_{(k)}$ of a large dimensional Gaussian random matrix. We find that this probability behaves for large $N$ as $\mathcal{P}[\lambda_{(k)}=x]\approx \exp\left(-\beta N^2 \psi(k/N,x)\right)$, where $\beta$ is the Dyson index of the ensemble. The rate function $\psi(c,x)$, computed explicitly as a function of $x$ in terms of the intensive label $c=k/N$, has a quadratic behavior modulated by a weak logarithmic singularity at its minimum. This is shown to be related to phase transitions in the associated Coulomb gas problem. The connection with statistics of extreme eigenvalues of random matrices is also elucidated.