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

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).

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

Letters in Mathematical Physics ◽

10.1007/bf00398398 ◽

1992 ◽

Vol 25 (4) ◽

pp. 259-265 ◽

Cited By ~ 20

Author(s):

L. A. Pastur

Keyword(s):

Random Matrices ◽

Level Spacing ◽

Spacing Distribution ◽

Level Spacing Distribution

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

Zeitschrift für Naturforschung A ◽

10.1515/zna-2007-0902 ◽

2007 ◽

Vol 62 (9) ◽

pp. 471-482 ◽

Cited By ~ 3

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.

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Influence of chemical disorder on the electronic level spacing distribution of the Ag5083 nanoparticle: A tight-binding study

Physica B Condensed Matter ◽

10.1016/j.physb.2012.12.027 ◽

2013 ◽

Vol 412 ◽

pp. 122-125 ◽

Cited By ~ 6

Author(s):

L.R. Medrano ◽

C.V. Landauro

Keyword(s):

Tight Binding ◽

Binding Study ◽

Level Spacing ◽

Electronic Level ◽

Spacing Distribution ◽

Chemical Disorder ◽

Level Spacing Distribution

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

Journal of the Physical Society of Japan ◽

10.1143/jpsj.62.2762 ◽

1993 ◽

Vol 62 (8) ◽

pp. 2762-2772 ◽

Cited By ~ 14

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

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Jensen polynomials for the Riemann zeta function and other sequences

Proceedings of the National Academy of Sciences ◽

10.1073/pnas.1902572116 ◽

2019 ◽

Vol 116 (23) ◽

pp. 11103-11110 ◽

Cited By ~ 9

Author(s):

Michael Griffin ◽

Ken Ono ◽

Larry Rolen ◽

Don Zagier

Keyword(s):

Partition Function ◽

Zeta Function ◽

Riemann Zeta Function ◽

General Theorem ◽

Hermite Polynomials ◽

Gaussian Unitary Ensemble ◽

Random Matrix Model ◽

The Riemann Zeta Function ◽

Riemann Zeta ◽

Unitary Ensemble

In 1927, Pólya proved that the Riemann hypothesis is equivalent to the hyperbolicity of Jensen polynomials for the Riemann zeta function ζ(s) at its point of symmetry. This hyperbolicity has been proved for degrees d≤3. We obtain an asymptotic formula for the central derivatives ζ(2n)(1/2) that is accurate to all orders, which allows us to prove the hyperbolicity of all but finitely many of the Jensen polynomials of each degree. Moreover, we establish hyperbolicity for all d≤8. These results follow from a general theorem which models such polynomials by Hermite polynomials. In the case of the Riemann zeta function, this proves the Gaussian unitary ensemble random matrix model prediction in derivative aspect. The general theorem also allows us to prove a conjecture of Chen, Jia, and Wang on the partition function.

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