Gaussian Unitary Ensemble with Boundary Spectrum Singularity and σ-Form of the Painlevé II Equation

2017 ◽  
Vol 140 (2) ◽  
pp. 221-251 ◽  
Author(s):  
Xiao-Bo Wu ◽  
Shuai-Xia Xu ◽  
Yu-Qiu Zhao
Keyword(s):  
Gaussian Unitary Ensemble ◽  
Boundary Spectrum ◽  
Painlevé Ii ◽  
Unitary Ensemble

scholarly journals Jensen polynomials for the Riemann zeta function and other sequences

2019 ◽  
Vol 116 (23) ◽  
pp. 11103-11110 ◽  
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.

scholarly journals Additive matrix convolutions of Pólya ensembles and polynomial ensembles

2019 ◽  
Vol 09 (04) ◽  
pp. 2150002
Author(s):  
Mario Kieburg
Keyword(s):  
Unit Element ◽  
Frequency Function ◽  
Semi Group ◽  
Orthogonal Functions ◽  
Toeplitz Determinants ◽  
Gaussian Unitary Ensemble ◽  
Finite Matrix ◽  
Transformation Formulas ◽  
Fixed Matrix ◽  
Unitary Ensemble

Recently, subclasses of polynomial ensembles for additive and multiplicative matrix convolutions were identified which were called Pólya ensembles (or polynomial ensembles of derivative type). Those ensembles are closed under the respective convolutions and, thus, build a semi-group when adding by hand a unit element. They even have a semi-group action on the polynomial ensembles. Moreover, in several works transformations of the bi-orthogonal functions and kernels of a given polynomial ensemble were derived when performing an additive or multiplicative matrix convolution with particular Pólya ensembles. For the multiplicative matrix convolution on the complex square matrices the transformations were even done for general Pólya ensembles. In the present work, we generalize these results to the additive convolution on Hermitian matrices, on Hermitian anti-symmetric matrices, on Hermitian anti-self-dual matrices and on rectangular complex matrices. For this purpose, we derive the bi-orthogonal functions and the corresponding kernel for a general Pólya ensemble which was not done before. With the help of these results, we find transformation formulas for the convolution with a fixed matrix or a random matrix drawn from a general polynomial ensemble. As an example, we consider Pólya ensembles with an associated weight which is a Pólya frequency function of infinite order. But we also explicitly evaluate the Gaussian unitary ensemble as well as the complex Laguerre (aka Wishart, Ginibre or chiral Gaussian unitary) ensemble. All results hold for finite matrix dimension. Furthermore, we derive a recursive relation between Toeplitz determinants which appears as a by-product of our results.

Generalization of Brody distribution for statistical investigation

2014 ◽  
Vol 03 (04) ◽  
pp. 1450017 ◽  
Author(s):  
H. Sabri ◽  
Sh. S. Hashemi ◽  
B. R. Maleki ◽  
M. A. Jafarizadeh
Keyword(s):  
Nearest Neighbor ◽  
Matrix Theory ◽  
Likelihood Estimation ◽  
Statistical Investigation ◽  
Distribution Functions ◽  
Estimation Methods ◽  
General Distribution ◽  
Gaussian Unitary Ensemble ◽  
Gaussian Orthogonal Ensemble ◽  
Unitary Ensemble

In this paper, Brody distribution is generalized to explore the Poisson, Gaussian Orthogonal Ensemble and Gaussian Unitary Ensemble limits of Random Matrix Theory in the nearest neighbor spacing statistic framework. Parameters of new distribution are extracted via Maximum Likelihood Estimation technique for different sequences. This general distribution suggests more exact results in comparison with the results of other estimation methods and distribution functions.

scholarly journals Asymptotic Independence in the Spectrum of the Gaussian Unitary Ensemble

2010 ◽  
Vol 15 (0) ◽  
pp. 376-395 ◽  
Author(s):  
Pascal Bianchi ◽  
Mérouane Debbah ◽  
Jamal Najim
Keyword(s):  
Asymptotic Independence ◽  
Gaussian Unitary Ensemble ◽  
Unitary Ensemble

scholarly journals Replica approach in random matrix theory

2018 ◽  
Author(s):  
Thomas Guhr
Keyword(s):  
Partition Function ◽  
Random Matrix Theory ◽  
Random Matrix ◽  
Matrix Theory ◽  
Field Theories ◽  
Partition Functions ◽  
Gaussian Unitary Ensemble ◽  
Integrable Theory ◽  
Unitary Ensemble ◽  
Replica Approach

This article examines the replica method in random matrix theory (RMT), with particular emphasis on recently discovered integrability of zero-dimensional replica field theories. It first provides an overview of both fermionic and bosonic versions of the replica limit, along with its trickery, before discussing early heuristic treatments of zero-dimensional replica field theories, with the goal of advocating an exact approach to replicas. The latter is presented in two elaborations: by viewing the β = 2 replica partition function as the Toda lattice and by embedding the replica partition function into a more general theory of τ functions. The density of eigenvalues in the Gaussian Unitary Ensemble (GUE) and the saddle point approach to replica field theories are also considered. The article concludes by describing an integrable theory of replicas that offers an alternative way of treating replica partition functions.

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