2014/04/30 by Ricardo Marino, Satya N. Majumdar, Grégory Schehr +1 · 1 citation
Physics and Astronomy · Mathematics · #cond-mat.stat-mech #math-ph #math.MP
paper · pdf · doi:10.1103/physrevlett.112.254101
published as Phys. Rev. Lett. 112, 254101 (2014) · 5 pag., 3 fig, published version
arxiv created 2014/06/27 · arxiv updated 2014/06/30
We consider N× N Gaussian random matrices, whose average density of eigenvalues has the Wigner semi-circle form over [-√(2),√(2)]. For such matrices, using a Coulomb gas technique, we compute the large N behavior of the probability P\scriptscriptstyle N,L(NL) that NL eigenvalues lie within the box [-L,L]. This probability scales as P\scriptscriptstyle N,L(NL=κL N)≈exp(-β N2 ψL(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-1≪ L<√(2) (bulk), (ii) L∼√(2) on a scale of O(N^-2/3) (edge) and (iii) L > √(2) (tail). We find a dramatic non-monotonic behavior of the number variance VN(L) as a function of L: after a logarithmic growth ∝ ln (N L) in the bulk (when L ∼ \cal O(1/N)), VN(L) decreases abruptly as L approaches the edge of the semi-circle before it decays as a stretched exponential for L > √(2). This "drop-off" of VN(L) at the edge is described by a scaling function Vβ which 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.