On the universality of the level spacing distribution for some ensembles of random matrices

1992 ◽  
Vol 25 (4) ◽  
pp. 259-265 ◽  
Author(s):  
L. A. Pastur
Keyword(s):  
Random Matrices ◽  
Level Spacing ◽  
Spacing Distribution ◽  
Level Spacing Distribution

scholarly journals Some Generic Properties of Level Spacing Distributions of 2D Real Random Matrices

2007 ◽  
Vol 62 (9) ◽  
pp. 471-482 ◽  
Author(s):  
Siegfried Grossmann ◽  
Marko Robnik
Keyword(s):  
Distribution Function ◽  
Random Matrices ◽  
Power Law ◽  
Preceding Paper ◽  
Matrix Elements ◽  
Level Spacing ◽  
Generic Properties ◽  
Spacing Distribution ◽  
The Matrix ◽  
Level Spacing Distribution

We study the level spacing distribution P(S) of 2D real random matrices both symmetric as well as general, non-symmetric. In the general case we restrict ourselves to Gaussian distributed matrix elements, but different widths of the various matrix elements are admitted. The following results are obtained: An explicit exact formula for P(S) is derived and its behaviour close to S = 0 is studied analytically, showing that there is linear level repulsion, unless there are additional constraints for the probability distribution of the matrix elements. The constraint of having only positive or only negative but otherwise arbitrary non-diagonal elements leads to quadratic level repulsion with logarithmic corrections. These findings detail and extend our previous results already published in a preceding paper. For symmetric real 2D matrices also other, non-Gaussian statistical distributions are considered. In this case we show for arbitrary statistical distribution of the diagonal and non-diagonal elements that the level repulsion exponent ρ is always ρ = 1, provided the distribution function of the matrix elements is regular at zero value. If the distribution function of the matrix elements is a singular (but still integrable) power law near zero value of S, the level spacing distribution P(S) is a fractional exponent power law at small S. The tail of P(S) depends on further details of the matrix element statistics. We explicitly work out four cases: the uniform (box) distribution, the Cauchy-Lorentz distribution, the exponential distribution and, as an example for a singular distribution, the power law distribution for P(S) near zero value times an exponential tail.

scholarly journals A PSEUDO-UNITARY ENSEMBLE OF RANDOM MATRICES, PT-SYMMETRY AND THE RIEMANN HYPOTHESIS

2006 ◽  
Vol 21 (04) ◽  
pp. 331-338 ◽  
Author(s):  
ZAFAR AHMED ◽  
SUDHIR R. JAIN
Keyword(s):  
Random Matrices ◽  
Riemann Hypothesis ◽  
Level Spacing ◽  
Gaussian Unitary Ensemble ◽  
Spacing Distribution ◽  
Hamiltonian Connected ◽  
Spectral Interpretation ◽  
Riemann Zeta ◽  
Unitary Ensemble ◽  
Level Spacing Distribution

An ensemble of 2×2 pseudo-Hermitian random matrices is constructed that possesses real eigenvalues with level-spacing distribution exactly as for the Gaussian unitary ensemble found by Wigner. By a re-interpretation of Connes' spectral interpretation of the zeros of Riemann zeta function, we propose to enlarge the scope of search of the Hamiltonian connected with the celebrated Riemann hypothesis by suggesting that the Hamiltonian could also be PT-symmetric (or pseudo-Hermitian).

Level Spacing Distribution and Δ3-Statistics of Two Dimensional Disordered Electrons in Strong Magnetic Field

1993 ◽  
Vol 62 (8) ◽  
pp. 2762-2772 ◽  
Author(s):  
Yoshiyuki Ono ◽  
Hiroyuki Kuwano ◽  
Keith Slevin ◽  
Tomi Ohtsuki ◽  
and Bernhard Kramer
Keyword(s):  
Magnetic Field ◽  
Strong Magnetic Field ◽  
Two Dimensional ◽  
Level Spacing ◽  
Spacing Distribution ◽  
Level Spacing Distribution

scholarly journals Level-spacing distribution of a singular billiard

1993 ◽  
Vol 47 (6) ◽  
pp. R3822-R3825 ◽  
Author(s):  
T. Shigehara ◽  
N. Yoshinaga ◽  
Taksu Cheon ◽  
T. Mizusaki
Keyword(s):  
Level Spacing ◽  
Spacing Distribution ◽  
Level Spacing Distribution

Numerical demonstration of the Berry-Robnik level spacing distribution

1994 ◽  
Vol 27 (13) ◽  
pp. L459-L466 ◽  
Author(s):  
T Prosen ◽  
M Robnik
Keyword(s):  
Level Spacing ◽  
Spacing Distribution ◽  
Level Spacing Distribution

Level Spacing Distribution and Avoided Crossing in Quantum Chaos

1987 ◽  
Vol 56 (8) ◽  
pp. 2641-2652 ◽  
Author(s):  
Akira Shudo ◽  
Nobuhiko Saitô
Keyword(s):  
Quantum Chaos ◽  
Level Spacing ◽  
Spacing Distribution ◽  
Avoided Crossing ◽  
Level Spacing Distribution

Low energy level spacing distribution in the atomic table ensemble

1980 ◽  
Vol 80 (1) ◽  
pp. 14-19 ◽  
Author(s):  
V.K.B. Kota ◽  
V. Potbhare
Keyword(s):  
Energy Level ◽  
Low Energy ◽  
Level Spacing ◽  
Spacing Distribution ◽  
Level Spacing Distribution
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