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Phase transitions and edge scaling of number variance in Gaussian random matrices

We consider $N\times N$ Gaussian random matrices, whose average density of eigenvalues has the Wigner semi-circle form over $[-\sqrt{2},\sqrt{2}]$. For such matrices, using a Coulomb gas technique, we compute the large $N$ behavior of the probability $\mathcal{P}_{\scriptscriptstyle N,L}(N_L)$ that $N_L$ eigenvalues lie within the box $[-L,L]$. This probability scales as $\mathcal{P}_{\scriptscriptstyle N,L}(N_L=κ_L N)\approx\exp\left(-β N^2 ψ_L(κ_L)\right)$, where $β$ is the Dyson index of the ensemble and $ψ_L(κ_L)$ is a $β$-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 non-monotonic behavior of the number variance $V_N(L)$ as a function of $L$: after a logarithmic growth $\propto \ln (N L)$ in the bulk (when $L \sim {\cal O}(1/N)$), $V_N(L)$ decreases abruptly as $L$ approaches the edge of the semi-circle before it decays as a stretched exponential for $L > \sqrt{2}$. This "drop-off" of $V_N(L)$ at the edge is described by a scaling function $\tilde V_β$ which smoothly interpolates between the bulk (i) and the tail (iii). For $β= 2$ we compute $\tilde V_2$ explicitly in terms of the Airy kernel. These analytical results, verified by numerical simulations, directly provide for $β=2$ the full statistics of particle-number fluctuations at zero temperature of 1d spinless fermions in a harmonic trap.