Phase Transitions and Edge Scaling of Number Variance in Gaussian Random Matrices

We consider N x N Gaussian random matrices, whose average density of eigenvalues has the Wigner semicircle form over [-root 2, root 2]. For such matrices, using a Coulomb gas technique, we compute the large N behavior of the probability P-N,P-L(N-L) that N-L eigenvalues lie within the box [-L, L]. This probability scales as P-N,P-L(N-L = kappa N-L) approximate to exp (-beta N-2 psi(L)(kappa(L))), 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 << L < root 2(bulk), (ii) L similar to root 2 on a scale of O(N-2/3) (edge), and (iii) L > root 2 (tail). We find a dramatic nonmonotonic behavior of the number variance V-N(L) as a function of L: after a logarithmic growth proportional to ln(NL) in the bulk (when L similar to 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 > root 2. This dropoff of V-N(L) at the edge is described by a scaling function (V) over tilde (beta) that smoothly interpolates between the bulk (i) and the tail (iii). For beta = 2 we compute (V) over tilde (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.