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

We consider N×N Gaussian random matrices, whose average density of eigenvalues has the Wigner semicircle form over [-√2,√2]. For such matrices, using a Coulomb gas technique, we compute the large N behavior of the probability PN,L(NL) that NL eigenvalues lie within the box [-L,L]. This probability scales as PN,L(NL=κLN)≈exp(-βN²ψL(κL)), where β is the Dyson index of the ensemble and ψL(κL) is a β-independent rate function that we compute exactly. We identify three regimes as L is varied: (i) N^-1L<√2 (bulk), (ii) L~√2 on a scale of O(N^-2/3) (edge), and (iii) L>√2 (tail). We find a dramatic nonmonotonic behavior of the number variance VN(L) as a function of L: after a logarithmic growth ∝ln(NL) in the bulk (when L~O(1/N)), VN(L) decreases abruptly as L approaches the edge of the semicircle before it decays as a stretched exponential for L>√2. This ``dropoff'' of VN(L) at the edge is described by a scaling function V_β that smoothly interpolates between the bulk (i) and the tail (iii). For β=2 we compute V₂ 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.