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 spinless fermions (or, equivalently, hard bosons) in a harmonic trap at zero temperature, including its large deviation tails. The problem amounts to computing the probability distribution of the $k\mathrm{th}$ smallest eigenvalue ${\lambda }_{\left(k\right)}$ of a large dimensional Gaussian random matrix. We find that this probability behaves for large $N$ as $\mathcal{P}[{\lambda }_{\left(k\right)}=x]\approx exp[-\beta N^2\psi (k/N,x)]$, 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 and order stastistics of random matrices is also discussed. We find that, as a function of $c$ and keeping the value of $x$ fixed, the rate function $\psi (c,x)$ describes the statistics of the shifted index number, generalizing known results on its typical fluctuations; as a function of $x$ and keeping the fraction $c=k/N$ fixed, the rate function $\psi (c,x)$ also describes the statistics of the $k\mathrm{th}$ eigenvalue in the bulk, generalizing as well the results on its typical fluctuations. Moreover, for $k=1$ (respectively, for $k=N$), the rate function captures both the fluctuations to the left and to the right of the typical value of ${\lambda }_{\left(1\right)}$ (respectively, of ${\lambda }_{\left(N\right)}$).

Figure 1

Plot of the rate function $\psi (c,x)$ as a function of $c\in [0,1]$ (orange) and $x\in \mathbb{R}$ (blue), yielding the large deviation decay of $N_x$ and ${\lambda }_{\left(k\right)}$, respectively. The rate function is zero along the red line in the plane [Eq. (7)], corresponding to the unperturbed (semicircle) phase $\text{III}$ in Fig. 2. The two curves have been rescaled so that they are both visible in the same figure.

Figure 2

Top: Possible equilibrium phases of the Coulomb gas (14). Bottom: Corresponding regions in the $(c,x)$ strip. Phase $\text{III}$ corresponds to the unperturbed semicircle law ${\rho }_{\mathrm{sc}}$. Phases $\text{I}$ and $\text{II}$ are single support and correspond to having all ($c=1$ and $x<\sqrt{2}$) or none ($c=0$ and $x>-\sqrt{2}$) of the eigenvalues to the left of $x$. The color code corresponds to regions where the action (13) at the saddle point $E\left[{\rho }_{c,x}^{☆}\right]$ is larger (darker) or smaller (lighter).