Abstract
We propose a plasma model for spectral statistics displaying level repulsion without long-range spectral rigidity, i.e. statistics intermediate between random matrix and Poisson statistics similar to the ones found numerically at the critical point of the Anderson metal-insulator transition in disordered systems and in certain dynamical systems. The model emerges from Dysons one-dimensional gas corresponding to the eigenvalue distribution of the classical random matrix ensembles by restricting the logarithmic pair interaction to a finite number k of nearest neighbors. We calculate analytically the spacing distributions and the two-level statistics. In particular we show that the number variance has the asymptotic form ∑2(L) ∼ χL for large L and the nearest-neighbor distribution decreases exponentially when s → ∞, P(s) ∼ exp(-Λs) with A = 1/χ = kβ + 1, where β is the inverse temperature of the gas (β = 1, 2 and 4 for the orthogonal, unitary and symplectic symmetry class respectively). In the simplest case of k = β= 1, the model leads to the so-called Semi-Poisson statistics characterized by particular simple correlation functions e.g. P(S) = 4s exp(-2s). Furthermore we investigate the spectral statistics of several pseudointegrable quantum billiards numerically and compare them to the Semi-Poisson statistics.
Original language | English |
---|---|
Pages (from-to) | 121-132 |
Number of pages | 12 |
Journal | European Physical Journal B |
Volume | 19 |
Issue number | 1 |
DOIs | |
State | Published - 1 Jan 2001 |
Externally published | Yes |
Keywords
- 03.65.Sq Semiclassical theories and applications
- 05.40.-a Fluctuation phenomena, random processes, noise and Brownian motion
- 05.45.-a Nonlinear dynamics and nonlinear dynamical systems