Spectral statistics of unitary ensembles

2018 ◽  
Author(s):  
Freeman Dyson
Keyword(s):  
Orthogonal Polynomials ◽  
Polynomial Method ◽  
Real Numbers ◽  
Fredholm Determinants ◽  
Point Correlation ◽  
Point Correlation Function ◽  
Spectral Statistics ◽  
Matrix Ensembles ◽  
Gap Probability ◽  
Computing Correlation

This article focuses on the use of the orthogonal polynomial method for computing correlation functions, cluster functions, gap probability, Janossy density, and spacing distributions for the eigenvalues of matrix ensembles with unitary-invariant probability law. It first considers the classical families of orthogonal polynomials (Hermite, Laguerre, and Jacobi) and some corresponding unitary ensembles before discussing the statistical properties of N-tuples of real numbers. It then reviews the definitions of basic statistical quantities and demonstrates how their distributions can be made explicit in terms of orthogonal polynomials. It also describes the k-point correlation function, Fredholm determinants of finite-rank kernels, and resolvent kernels.

scholarly journals Bifurcation and anomalous spectral accumulation in an oval billiard

2019 ◽  
Vol 2019 (8) ◽  
Author(s):  
Hironori Makino
Keyword(s):  
Dynamical System ◽  
Phase Space ◽  
Limited Region ◽  
Classical Dynamical System ◽  
Classical Trajectories ◽  
Point Correlation ◽  
Point Correlation Function ◽  
Spectral Statistics ◽  
Math Phys ◽  
Good Agreement

Abstract The spectral statistics of a quantum oval billiard whose classical dynamical system shows bifurcations is numerically investigated in terms of the two-point correlation function (TPCF), which is defined as the probability density of finding two levels at a specific energy interval. The eigenenergy levels at the bifurcation point are found to show anomalous accumulation, which is observed as a periodic spike oscillation of the TPCF. We analyzed the eigenfunctions localizing onto the various classical trajectories in the phase space and found that the oscillation is supplied from a limited region in the phase space that contains the bifurcating orbit. We also show that the period of the oscillation is in good agreement with the period of a contribution from the bifurcating orbit to the semiclassical TPCF obtained by Gutzwiller’s trace formula [J. Math. Phys. 12, 343 (1971)].

scholarly journals Spectral statistics of permutation matrices

2014 ◽  
Vol 372 (2007) ◽  
pp. 20120508 ◽  
Author(s):  
Idan Oren ◽  
Uzy Smilansky
Keyword(s):  
Number Theory ◽  
Point Of View ◽  
Spectral Form ◽  
Permutation Matrices ◽  
Divisor Functions ◽  
Point Correlation ◽  
Point Correlation Function ◽  
The Subject ◽  
Spectral Statistics ◽  
The Mean

We compute the mean two-point spectral form factor and the spectral number variance for permutation matrices of large order. The two-point correlation function is expressed in terms of generalized divisor functions, which are frequently discussed in number theory. Using classical results from number theory and casting them in a convenient form, we derive expressions which include the leading and next to leading terms in the asymptotic expansion, thus providing a new point of view on the subject, and improving some known results.

scholarly journals From correlation functions to event shapes in QCD

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
D. Chicherin ◽  
J. M. Henn ◽  
E. Sokatchev ◽  
K. Yan
Keyword(s):  
Correlation Function ◽  
Correlation Functions ◽  
Event Shape ◽  
Proof Of Concept ◽  
Yang Mills ◽  
Point Correlation ◽  
Point Correlation Function ◽  
Event Shapes ◽  
A Charge ◽  
Conserved Currents

Abstract We present a method for calculating event shapes in QCD based on correlation functions of conserved currents. The method has been previously applied to the maximally supersymmetric Yang-Mills theory, but we demonstrate that supersymmetry is not essential. As a proof of concept, we consider the simplest example of a charge-charge correlation at one loop (leading order). We compute the correlation function of four electromagnetic currents and explain in detail the steps needed to extract the event shape from it. The result is compared to the standard amplitude calculation. The explicit four-point correlation function may also be of interest for the CFT community.

scholarly journals A two-point correlation function for Galactic halo stars

2011 ◽  
Vol 417 (3) ◽  
pp. 2206-2215 ◽  
Author(s):  
A. P. Cooper ◽  
S. Cole ◽  
C. S. Frenk ◽  
A. Helmi
Keyword(s):  
Correlation Function ◽  
Galactic Halo ◽  
Halo Stars ◽  
Point Correlation ◽  
Point Correlation Function

scholarly journals TOWARDS THE PROOF OF AGT-W RELATION: TODA 3-POINT FUNCTION AND ${\mathcal W}_{1+\infty}$ ALGEBRA

2013 ◽  
Vol 21 ◽  
pp. 138-139
Author(s):  
SHOTARO SHIBA
Keyword(s):  
Field Theory ◽  
Correlation Function ◽  
Point Function ◽  
Gauge Groups ◽  
Promising Direction ◽  
Quiver Gauge Theory ◽  
Conformal Theory ◽  
Conformal Field ◽  
Point Correlation ◽  
Point Correlation Function

The AGT-W relation is a conjecture of the nontrivial duality between 4-dim quiver gauge theory and 2-dim conformal field theory. We verify a part of this conjecture for all the cases of quiver gauge groups by studying on the property of 3-point correlation function of conformal theory. We also mention the relation to [Formula: see text] algebra as one of the promising direction towards the proof of the remaining part.

scholarly journals A general treatment of snow microstructure exemplified by an improved relation for the thermal conductivity

2012 ◽  
Vol 6 (6) ◽  
pp. 4673-4693 ◽  
Author(s):  
H. Löwe ◽  
F. Riche ◽  
M. Schneebeli
Keyword(s):  
Thermal Conductivity ◽  
Correlation Function ◽  
Pore Space ◽  
Anisotropy Parameter ◽  
Micro Computed Tomography ◽  
Data Set ◽  
Microstructural Parameters ◽  
Point Correlation ◽  
Point Correlation Function ◽  
Snow Microstructure

Abstract. Finding relevant microstructural parameters beyond the density is a longstanding problem which hinders the formulation of accurate parametrizations of physical properties of snow. Towards a remedy we address the effective thermal conductivity tensor of snow via known anisotropic, second-order bounds. The bound provides an explicit expression for the thermal conductivity and predicts the relevance of a microstructural anisotropy parameter Q which is given by an integral over the two-point correlation function and unambiguously defined for arbitrary snow structures. For validation we compiled a comprehensive data set of 167 snow samples. The set comprises individual samples of various snow types and entire time series of metamorphism experiments under isothermal and temperature gradient conditions. All samples were digitally reconstructed by micro-computed tomography to perform microstructure-based simulations of heat transport. The incorporation of anisotropy via Q considerably reduces the root mean square error over the usual density-based parametrization. The systematic quantification of anisotropy via the two-point correlation function suggests a generalizable route to incorporate microstructure into snowpack models. We indicate the inter-relation of the conductivity to other properties and outline a potential impact of Q on dielectric constant, permeability and adsorption rate of diffusing species in the pore space.

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