Abstract
We consider Gaussian random matrices, whose average density of eigenvalues has the Wigner semicircle form over . For such matrices, using a Coulomb gas technique, we compute the large behavior of the probability that eigenvalues lie within the box . This probability scales as , where is the Dyson index of the ensemble and is a -independent rate function that we compute exactly. We identify three regimes as is varied: (i) (bulk), (ii) on a scale of (edge), and (iii) (tail). We find a dramatic nonmonotonic behavior of the number variance as a function of : after a logarithmic growth in the bulk (when ), decreases abruptly as approaches the edge of the semicircle before it decays as a stretched exponential for . This “dropoff” of at the edge is described by a scaling function that smoothly interpolates between the bulk (i) and the tail (iii). For we compute explicitly in terms of the Airy kernel. These analytical results, verified by numerical simulations, directly provide for 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
© 2014 American Physical Society