scholarly journals Correlation Functions, Universal Ratios and Goldstone Mode Singularities in n-Vector Models

2014 ◽  
Vol 15 (5) ◽  
pp. 1407-1430
Author(s):  
J. Kaupužs ◽  
R. V. N. Melnik ◽  
J. Rimšāns
Keyword(s):  
Correlation Functions ◽  
Gaussian Approximation ◽  
Spontaneous Magnetization ◽  
Standard Theory ◽  
Goldstone Mode ◽  
Simulation Data ◽  
Point Correlation ◽  
Gaussian Theory ◽  
Non Gaussian ◽  
N Vector

AbstractCorrelation functions in the (n) models below the critical temperature are considered. Based on Monte Carlo (MC) data, we confirm the fact stated earlier by Engels and Vogt, that the transverse two-plane correlation function of the (4) model for lattice sizes about L = 120 and small external fields h is very well described by a Gaussian approximation. However, we show that fits of not lower quality are provided by certain non-Gaussian approximation. We have also tested larger lattice sizes, up to L = 512. The Fourier-transformed transverse and longitudinal two-point correlation functions have Goldstone mode singularities in the thermodynamic limit at k → 0 and h = +0, i.e., G⊥ (k) ≃ ak–λ⊥ and G‖(k)≃bk–λ‖, respectively. Here a and b are the amplitudes, k = |k| is the magnitude of the wave vector k. The exponents λᚆ, λ‖ and the ratio bM2/a2, where M is the spontaneous magnetization, are universal according to the GFD (grouping of Feynman diagrams) approach. Here we find that the universality follows also from the standard (Gaussian) theory, yielding bM2/a2=(n−1)/16. Our MC estimates of this ratio are 0.06±0.01 for n=2, 0.17±0.01 for n = 4 and 0.498±0.010 for n = 10. According to these and our earlier MC results, the asymptotic behavior and Goldstone mode singularities are not exactly described by the standard theory. This is expected from the GFD theory. We have found appropriate analytic approximations for G⊥(k) and G‖(k), well fitting the simulation data for small k. We have used them to test the Patashinski-Pokrovski relation and have found that it holds approximately.

scholarly journals Non-Gaussianity in the weak lensing correlation function likelihood – implications for cosmological parameter biases

2020 ◽  
Vol 499 (2) ◽  
pp. 2977-2993
Author(s):  
Chien-Hao Lin ◽  
Joachim Harnois-Déraps ◽  
Tim Eifler ◽  
Taylor Pospisil ◽  
Rachel Mandelbaum ◽  
...  
Keyword(s):  
Correlation Functions ◽  
Principal Component ◽  
Cosmological Parameter ◽  
Weak Lensing ◽  
Likelihood Model ◽  
Point Correlation ◽  
Parametric Functions ◽  
Non Gaussian ◽  
Parameter Constraints ◽  
Skewness And Kurtosis

ABSTRACT We study the significance of non-Gaussianity in the likelihood of weak lensing shear two-point correlation functions, detecting significantly non-zero skewness and kurtosis in 1D marginal distributions of shear two-point correlation functions in simulated weak lensing data. We examine the implications in the context of future surveys, in particular LSST, with derivations of how the non-Gaussianity scales with survey area. We show that there is no significant bias in 1D posteriors of Ωm and σ8 due to the non-Gaussian likelihood distributions of shear correlations functions using the mock data (100 deg2). We also present a systematic approach to constructing approximate multivariate likelihoods with 1D parametric functions by assuming independence or more flexible non-parametric multivariate methods after decorrelating the data points using principal component analysis (PCA). While the use of PCA does not modify the non-Gaussianity of the multivariate likelihood, we find empirically that the 1D marginal sampling distributions of the PCA components exhibit less skewness and kurtosis than the original shear correlation functions. Modelling the likelihood with marginal parametric functions based on the assumption of independence between PCA components thus gives a lower limit for the biases. We further demonstrate that the difference in cosmological parameter constraints between the multivariate Gaussian likelihood model and more complex non-Gaussian likelihood models would be even smaller for an LSST-like survey. In addition, the PCA approach automatically serves as a data compression method, enabling the retention of the majority of the cosmological information while reducing the dimensionality of the data vector by a factor of ∼5.

scholarly journals THE FREE FIELD REPRESENTATION OF SU(3) CONFORMAL FIELD THEORY

1993 ◽  
Vol 08 (23) ◽  
pp. 4031-4053
Author(s):  
HOVIK D. TOOMASSIAN
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Correlation Functions ◽  
Free Field ◽  
Field Representation ◽  
Conformal Field ◽  
Point Correlation

The structure of the free field representation and some four-point correlation functions of the SU(3) conformal field theory are considered.

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 Towards a self-consistent analysis of the anisotropic galaxy two- and three-point correlation functions on large scales: application to mock galaxy catalogues

2020 ◽  
Author(s):  
Naonori S Sugiyama ◽  
Shun Saito ◽  
Florian Beutler ◽  
Hee-Jong Seo
Keyword(s):  
Correlation Functions ◽  
Hubble Parameter ◽  
Theoretical Models ◽  
Practical Method ◽  
Acoustic Oscillation ◽  
Nonlinear Damping ◽  
Joint Analysis ◽  
Angular Diameter Distance ◽  
Point Correlation ◽  
Parameter Constraints

Abstract We establish a practical method for the joint analysis of anisotropic galaxy two- and three-point correlation functions (2PCF and 3PCF) on the basis of the decomposition formalism of the 3PCF using tri-polar spherical harmonics. We perform such an analysis with MultiDark Patchy mock catalogues to demonstrate and understand the benefit of the anisotropic 3PCF. We focus on scales above 80 h−1 Mpc, and use information from the shape and the baryon acoustic oscillation (BAO) signals of the 2PCF and 3PCF. We also apply density field reconstruction to increase the signal-noise ratio of BAO in the 2PCF measurement, but not in the 3PCF measurement. In particular, we study in detail the constraints on the angular diameter distance and the Hubble parameter. We build a model of the bispectrum or 3PCF that includes the nonlinear damping of the BAO signal in redshift space. We carefully account for various uncertainties in our analysis including theoretical models of the 3PCF, window function corrections, biases in estimated parameters from the fiducial values, the number of mock realizations to estimate the covariance matrix, and bin size. The joint analysis of the 2PCF and 3PCF monopole and quadrupole components shows a $30\%$ and $20\%$ improvement in Hubble parameter constraints before and after reconstruction of the 2PCF measurements, respectively, compared to the 2PCF analysis alone. This study clearly shows that the anisotropic 3PCF increases cosmological information from galaxy surveys and encourages further development of the modeling of the 3PCF on smaller scales than we consider.

scholarly journals Correlation functions in finite temperature CFT and black hole singularities

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
D. Rodriguez-Gomez ◽  
J.G. Russo
Keyword(s):  
Black Hole ◽  
Event Horizon ◽  
Correlation Functions ◽  
Finite Temperature ◽  
Proper Time ◽  
Black Brane ◽  
Point Function ◽  
Black Hole Singularity ◽  
Point Correlation ◽  
Scaling Dimension

Abstract We compute thermal 2-point correlation functions in the black brane AdS5 background dual to 4d CFT’s at finite temperature for operators of large scaling dimension. We find a formula that matches the expected structure of the OPE. It exhibits an exponentiation property, whose origin we explain. We also compute the first correction to the two-point function due to graviton emission, which encodes the proper time from the event horizon to the black hole singularity.

scholarly journals Geometrical four-point functions in the two-dimensional critical Q-state Potts model: the interchiral conformal bootstrap

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Yifei He ◽  
Jesper Lykke Jacobsen ◽  
Hubert Saleur
Keyword(s):  
Potts Model ◽  
Correlation Functions ◽  
Liouville Theory ◽  
Analytical Determination ◽  
Conformal Weight ◽  
Two Dimensional ◽  
Conformal Blocks ◽  
Conformal Bootstrap ◽  
Point Correlation

Abstract Based on the spectrum identified in our earlier work [1], we numerically solve the bootstrap to determine four-point correlation functions of the geometrical connectivities in the Q-state Potts model. Crucial in our approach is the existence of “interchiral conformal blocks”, which arise from the degeneracy of fields with conformal weight hr,1, with r ∈ ℕ*, and are related to the underlying presence of the “interchiral algebra” introduced in [2]. We also find evidence for the existence of “renormalized” recursions, replacing those that follow from the degeneracy of the field $$ {\Phi}_{12}^D $$ Φ 12 D in Liouville theory, and obtain the first few such recursions in closed form. This hints at the possibility of the full analytical determination of correlation functions in this model.

scholarly journals Two point functions in defect CFTs

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Christopher P. Herzog ◽  
Abhay Shrestha
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Correlation Functions ◽  
Equations Of Motion ◽  
Physical Space ◽  
Arbitrary Spin ◽  
Momentum Tensor ◽  
Maxwell Theory ◽  
Conformal Field ◽  
Point Correlation

Abstract This paper is designed to be a practical tool for constructing and investigating two-point correlation functions in defect conformal field theory, directly in physical space, between any two bulk primaries or between a bulk primary and a defect primary, with arbitrary spin. Although geometrically elegant and ultimately a more powerful approach, the embedding space formalism gets rather cumbersome when dealing with mixed symmetry tensors, especially in the projection to physical space. The results in this paper provide an alternative method for studying two-point correlation functions for a generic d-dimensional conformal field theory with a flat p-dimensional defect and d − p = q co-dimensions. We tabulate some examples of correlation functions involving a conserved current, an energy momentum tensor and a Maxwell field strength, while analysing the constraints arising from conservation and the equations of motion. A method for obtaining bulk-to-defect correlators is also explained. Some explicit examples are considered: free scalar theory on ℝp× (ℝq/ℤ2) and a free four dimensional Maxwell theory on a wedge.

Adaptive and augmented nonlinear filters : theory and applications

2018 ◽  
Author(s):  
◽  
Tao Sun
Keyword(s):  
Kalman Filter ◽  
Mean Square Error ◽  
Gaussian Approximation ◽  
Adaptive Strategy ◽  
Nonlinear Filters ◽  
Nonlinear Transformation ◽  
Mean Square ◽  
Gaussian Filters ◽  
Non Gaussian ◽  
Gaussian Estimation

[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT AUTHOR'S REQUEST.] Nonlinear estimation and filtering have been intensively studied for decades since it has been widely used in engineering and science such as navigation, radar signal processing and target tracking systems. Because the posterior density function is not a Gaussian distribution, then the optimal solution is intractable. The nonlinear/non-Gaussian estimation problem is more challenging than the linear/Gaussian case, which has an optimal closed form solution, i.e. the celebrated Kalman filter. Many nonlinear filters including the extended Kalman filter, the unscented Kalman filter and the Gaussian-approximation filters, have been proposed to address nonlinear/non-Gaussian estimation problems in the past decades. Although the estimate yield by Gaussian-approximation filters such as cubature Kalman filters and Gaussian-Hermite quadrature filters is satisfied in many applications, there are two obvious drawbacks embedded in the use of Gaussian filters. On the one hand, with the increase of the quadrature points, much computational effort is devoted to approximate Gaussian integrals, which is not worthy sometimes. On the other hand, by the use of the update rule, the estimate constrains to be a linear function of the observation. In this dissertation, we aim to address this two shortcoming associated with the conventional nonlinear filters. We propose two nonlinear filters in the dissertation. Based on an adaptive strategy, the first one tries to reduce the computation cost during filtering without sacrificing much accuracy, because when the system is close to be linear, the lower level Gaussian quadrature filter is sufficient to provide accurate estimate. The adaptive strategy is used to evaluate the nonlinearity of the system at current time first and then utilize different quadrature rule for filtering. Another filter aims to modify the conventional update rule, i.e. the linear minimum mean square error (LMMSE) rule, to involve a nonlinear transformation of the observation, which is proven to be an efficient way to exploit more information from the original observation. According to the orthogonal property, we propose a novel approach to construct the nonlinear transformation systematically. The augmented nonlinear filter outperforms Gaussian filters and other conventional augmented filters in terms of the root mean square error and onsistency. Furthermore, we also extend the work to the more general case. The higher order moments can be utilized to construct the nonlinear transformation and in turn, the measurement space can be expand efficiently. Without the Gaussian assumption, the construction of the nonlinear transformation only demand the existence of a finite number of moments. Finally, the simulation results validate and demonstrate the superiority of the adaptive and augmented nonlinear filters.

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