We consider $N\times N$ Gaussian random matrices, whose average density of eigenvalues has the Wigner semicircle form over $[-\sqrt{2},\sqrt{2}]$. For such matrices, using a Coulomb gas technique, we compute the large $N$ behavior of the probability ${\mathcal{P}}_{N,L}(N_L)$ that $N_L$ eigenvalues lie within the box $[-L,L]$. This probability scales as ${\mathcal{P}}_{N,L}(N_L={\kappa }_{L}N)\approx \exp \mathbf{(}-\beta N^2{\psi }_L({\kappa }_L)\mathbf{)}$, where $\beta$ is the Dyson index of the ensemble and ${\psi }_L({\kappa }_L)$ is a $\beta$-independent rate function that we compute exactly. We identify three regimes as $L$ is varied: (i) $N^{-1}\ll L<\sqrt{2}$ (bulk), (ii) $L\sim \sqrt{2}$ on a scale of $\mathcal{O}(N^{-2/3})$ (edge), and (iii) $L>\sqrt{2}$ (tail). We find a dramatic nonmonotonic behavior of the number variance $V_N(L)$ as a function of $L$: after a logarithmic growth $\propto \ln (NL)$ in the bulk (when $L\sim \mathcal{O}(1/N)$), $V_N(L)$ decreases abruptly as $L$ approaches the edge of the semicircle before it decays as a stretched exponential for $L>\sqrt{2}$. This “dropoff” of $V_N(L)$ at the edge is described by a scaling function ${\tilde{V}}_{\beta }$ that smoothly interpolates between the bulk (i) and the tail (iii). For $\beta =2$ we compute ${\tilde{V}}_2$ explicitly in terms of the Airy kernel. These analytical results, verified by numerical simulations, directly provide for $\beta =2$ the full statistics of particle-number fluctuations at zero temperature of 1D spinless fermions in a harmonic trap.

• Received 2 April 2014

DOI:https://doi.org/10.1103/PhysRevLett.112.254101