Scaling Of Correlation Functions Research Articles

Metastability of Constant-Density Flocks

We study numerically the Toner-Tu field theory where the density field is maintained constant, a limit case of "Malthusian" flocks for which the asymptotic scaling of correlation functions in the ordered phase is known exactly. While we confirm these scaling laws, we also show that such constant-density flocks are metastable to the nucleation of a specific defect configuration, and are replaced by a globally disordered phase consisting of asters surrounded by shock lines that constantly evolves and remodels itself. We demonstrate that the main source of disorder lies along shock lines, rendering this active foam fundamentally different from the corresponding equilibrium system. We thus show that in the context of active matter also, a result obtained at all orders of perturbation theory can be superseded by nonperturbative effects, calling for a different approach.

Dynamics of non-Gaussian fluctuations in model A

Motivated by the experimental search for the QCD critical point we perform simulations of a stochastic field theory with purely relaxational dynamics (model A). We verify the expected dynamic scaling of correlation functions. Using a finite size scaling analysis we obtain the dynamic critical exponent $z=2.026(56)$. We investigate time dependent correlation functions of higher moments $M^n(t)$ of the order parameter $M(t)$ for $n=1,2,3,4$. We obtain dynamic scaling with the same critical exponent $z$ for all $n$, but the relaxation constant depends on $n$. We also study the relaxation of $M^n(t)$ after a quench, where the simulation is initialized in the high temperature phase, and the dynamics is studied at the critical temperature $T_c$. We find that the evolution does not follow simple scaling with the dynamic exponent $z$, and that it involves an early time rise followed by late stage relaxation.

Operator Scaling Dimensions and Multifractality at Measurement-Induced Transitions.

Repeated local measurements of quantum many-body systems can induce a phase transition in their entanglement structure. These measurement-induced phase transitions (MIPTs) have been studied for various types of dynamics, yet most cases yield quantitatively similar critical exponents, making it unclear how many distinct universality classes are present. Here, we probe the properties of the conformal field theories governing these MIPTs using a numerical transfer-matrix method, which allows us to extract the effective central charge, as well as the first few low-lying scaling dimensions of operators at these critical points for (1+1)-dimensional systems. Our results provide convincing evidence that the generic and Clifford MIPTs for qubits lie in different universality classes and that both are distinct from the percolation transition for qudits in the limit of large on-site Hilbert space dimension. For the generic case, we find strong evidence of multifractal scaling of correlation functions at the critical point, reflected in a continuous spectrum of scaling dimensions.

Anomalous scaling in the Kazantsev-Kraichnan model with finite time correlations: two-loop renormalization group analysis of relevant composite operators

The field theoretic renormalization group technique together with the operator product expansion in the second order of the perturbation theory (in the two-loop approximation) is used for the investigation of the influence of the finite time correlations of the velocity field on the anomalous dimensions of the leading set of composite operators, which drive the anomalous scaling of correlation functions of a weak magnetic field in the framework of the kinematic Kazantsev–Kraichnan model in the presence of a large scale anisotropy. The system of possible scaling regimes of the model is found and two important special limits of the model are briefly discussed. The general two-loop expressions for the anomalous and critical dimensions of the leading composite operators are found as functions of the spatial dimension d and of the renormalization group fixed point value of the parameter u, which drives the presence of the finite time correlations of the velocity field in the model. The anisotropic hierarchies among various anomalous dimensions are investigated and it is shown that, regardless of the fixed point value of the parameter u as well as regardless of the spatial dimension of the system, the leading role in the anomalous scaling properties of the model is played by the anomalous dimensions of the composite operators near the isotropic shell, in accordance with the Kolmogorov’s local isotropy restoration hypothesis. The properties of the anomalous dimensions of the leading composite operators in the Kazantsev–Kraichnan model with finite time correlations of the velocity field are compared to the properties of the corresponding anomalous dimensions of the composite operators relevant in the framework of the Kraichnan model of passively advected scalar field with finite time correlations. It is shown that, regardless of the fixed point value of the parameter u, the two-loop corrections to the anomalous dimensions are much more important in the framework of the Kazantsev–Kraichnan vector model than in the Kraichnan model of a passive scalar advection. At the same time, again regardless of the strength of time correlations, the two-loop values of the leading anomalous dimensions in the Kazantsev–Kraichnan model of the passive magnetic field are significantly more negative than the corresponding two-loop values of the relevant anomalous dimensions in the framework of the Kraichnan model. It means that the anomalous scaling of the correlation functions of the passive magnetic field, deep inside the inertial interval of the turbulent environment with finite time correlations of the velocity field, must be much more pronounced than in the case of the correlation functions of the passively advected scalar field.

Finite-size scaling of correlation functions in finite systems

We propose the finite-size scaling of correlation functions in finite systems near their critical points. At a distance ${\boldsymbol~r}$ in a $d$-dimensional finite system of size $L$, the correlation function can be written as the product of $|{\boldsymbol~r}|^{-(d-2+\eta)}$ and a finite-size scaling function of the variables ${\boldsymbol~r}/L$ and $tL^{1/\nu}$, where $t=(T-T_\text{c})/T_\text{c}$, $\eta$ is the critical exponent of correlation function, and $\nu$ is the critical exponent of correlation length. The correlation function only has a sigificant directional dependence when $|{\boldsymbol~r}|$ is compariable to $L$. We then confirm this finite-size scaling by calculating the correlation functions of the two-dimensional Ising model and the bond percolation in two-dimensional lattices using Monte Carlo simulations. We can use the finite-size scaling of the correlation function to determine the critical point and the critical exponent $\eta$.

Simultaneous influence of helicity and compressibility on anomalous scaling of the magnetic field in the Kazantsev-Kraichnan model.

Using the field theoretic renormalization group technique and the operator product expansion, the systematic investigation of the influence of the spatial parity violation on the anomalous scaling behavior of correlation functions of the weak passive magnetic field in the framework of the compressible Kazantsev-Kraichnan model with the presence of a large-scale anisotropy is performed up to the second order of the perturbation theory (two-loop approximation). The renormalization group analysis of the model is done and the two-loop explicit expressions for the anomalous and critical dimensions of the leading composite operators are found as functions of the helicity and compressibility parameters and their anisotropic hierarchies are discussed. It is shown that for arbitrary values of the helicity parameter and for physically acceptable (small enough) values of the compressibility parameter, the main role is played by the composite operators near the isotropic shell in accordance with the Kolmogorov's local isotropy restoration hypothesis. The anomalous dimensions of the relevant composite operators are then compared with the anomalous dimensions of the corresponding leading composite operators in the Kraichnan model of passively advected scalar field. The significant difference between these two sets of anomalous dimensions is discussed. The two-loop inertial-range scaling exponents of the single-time two-point correlation functions of the magnetic field are found and their dependence on the helicity and compressibility parameters is studied in detail. It is shown that while the presence of the helicity leads to more pronounced anomalous scaling for correlation functions of arbitrary order, the compressibility, in general, makes the anomalous scaling more pronounced in comparison to the incompressible case only for low-order correlation functions. The persistence of the anisotropy deep inside the inertial interval is investigated using the appropriate odd ratios of the correlation functions. It is shown that, in general, the persistence of the anisotropy is much more pronounced in the helical systems, while in the compressible turbulent environments this is true only for low-order odd ratios of the correlation functions.

Phase ordering kinetics of a nonequilibrium exciton-polariton condensate

We investigate the process of coarsening via annihilation of vortex-antivortex pairs, following the quench to the condensate phase in a nonresonantly pumped polariton system. We find that the late-time dynamics is an example of universal phase ordering kinetics, characterized by scaling of correlation functions in time. Depending on the parameters of the system, the evolution of the characteristic length scale L(t) can be the same as for the two-dimensional XY model, described by a power law with the dynamical exponent z \aprox 2 and a logarithmic correction, or z \aprox 1 which agrees with previous studies of conservative superfluids.

Precise clustering and density evolution of redMaPPer galaxy clusters versus MXXL simulation

We construct a large, redshift complete sample of distant galaxy clusters by correlating Sloan Digital Sky Survey (SDSS) Data Release 12 (DR12) redshifts with clusters identified with the red-sequence Matched-filter Probabilistic Percolation (redMaPPer) algorithm. Our spectroscopic completeness is 97% for ~ 7,000 clusters within the redMaPPer selection limit, $z \leqslant$ 0.325, so that our cluster correlation functions are much more precise than earlier work and not suppressed by photometric redshifts. We derive an accurate power-law mass-richness relation from the observed abundance with respect to the mass function from Millennium XXL (MXXL) simulation, adjusted to the Planck weighted cosmology. The number density of clusters is found to decline by 20% over the range 0.1 $< z <$ 0.3, in good agreement with the evolution predicted by MXXL. Our projected three-dimensional correlation function scales with richness, $\lambda$, rising from $r_0=$ 14 h$^{-1}$ Mpc at $\lambda\simeq$ 25, to $r_0=$ 22 h$^{-1}$ Mpc at $\lambda\simeq$ 60, with a gradient that matches MXXL when applying our mass-richness relation, whereas the observed amplitude of the correlation function at $\left<z\right>=$ 0.24 exceeds the MXXL prediction by 20% at the $\simeq$ 2.5$\sigma$ level. This tension cannot be blamed on spurious, randomly located clusters as this would reduce the correlation amplitude. Full consistency between the correlation function and the abundances is achievable for the pre-Planck values of $\sigma_8=$ 0.9, $\Omega_m=$ 0.25, and h = 0.73, matching the improved distance ladder estimate of the Hubble constant.

Intrinsic alignments of disc and elliptical galaxies in the MassiveBlack-II and Illustris simulations

We study the shapes and intrinsic alignments of disks and elliptical galaxies in the MassiveBlack-II (MBII) and Illustris cosmological hydrodynamic simulations, with volumes of $(100h^{-1}Mpc)^{3}$ and $(75h^{-1}Mpc)^{3}$ respectively. We find that simulated disk galaxies are more oblate in shape and more misaligned with the shape of their host dark matter subhalo when compared with ellipticals. The disk major axis is found to be oriented towards the location of nearby elliptical galaxies. We also find that the disks are thinner in MBII and misalignments with dark matter halo orientations are smaller in both disks and ellipticals when compared with Illustris. As a result, the intrinsic alignment correlation functions at fixed mass have a higher amplitude in MBII than in Illustris. Finally, at scales above $\sim 0.1$\hmpc, the intrinsic alignment two-point correlation functions for disk galaxies in both simulations are consistent with a null detection, unlike those for ellipticals. Despite significant differences in the treatments of hydrodynamics and baryonic physics in the simulations, we find that the $w_{\delta +}$ correlation function scales similarly with transverse separation. However, the less massive galaxies show different scale dependence in the ED correlation. This result indicates that, while hydrodynamic simulations are a promising tool to study intrinsic alignments, further study is needed to understand the impact of differences in the implementations of hydrodynamics and baryonic feedback.

AN OBSERVATIONAL SIGNAL OF THE VOID SHAPE CORRELATION AND ITS LINK TO THE COSMIC WEB

The shapes of cosmic voids are prone to distortions by the external tidal forces since their low-densities imply a lower internal resistance. This susceptibility of the void shapes to the tidal distortions makes them useful as an indicator of the large-scale tidal and density fields, despite the practical difficulty in defining them. Using the void catalog constructed by Pan et al. (2012) from the Seventh Data Release of the Sloan Digital Sky Survey (SDSS DR7), we detect a clear 4 sigma signal of spatial correlations of the void shapes on the scale of 20 Mpc/h and show that the signal is robust against the projection of the void shapes onto the plane of sky. By constructing a simple analytic model for the void shape correlation, within the framework of tidal torque theory, we demonstrate that the void shape correlation function scales linearly with the two-point correlation function of the linear density field. We also find a direct observational evidence for the cross-correlation of the void shapes with the large-scale velocity shear field that was linearly reconstructed by Lee et al. (2014) from the SDSS DR7. We discuss the possibility of using the void shape correlation function to break the degeneracy between the density parameter and the power spectrum amplitude and to independently constrain the neutrino mass as well.

Scaling functions in the square Ising model

We show and give the linear differential operators of order , for the integrals which appear in the two-point correlation scaling function of Ising model . The integrals are given in expansion around in the basis of the formal solutions of with transcendental combination coefficients. We find that the expression is a solution of the Painlevé VI equation in the scaling limit. Combinations of the (analytic at ) solutions of sum to . We show that the expression is the scaling limit of the correlation function and . The differential Galois groups of the factors occurring in the operators are given.

Conducting fixed points for inhomogeneous quantum wires: A conformally invariant boundary theory

Inhomogeneities and junctions in wires are natural sources of scattering, and hence resistance. A conducting fixed point usually requires an adiabatically smooth system. One notable exception is "healing", which has been predicted in systems with special symmetries, where the system is driven to the homogeneous fixed point. Here we present theoretical results for a different type of conducting fixed point which occurs in inhomogeneous wires with an abrupt jump in hopping and interaction strength. We show that it is always possible to tune the system to an unstable conducting fixed point which does not correspond to translational invariance. We analyze the temperature scaling of correlation functions at and near this fixed point and show that two distinct boundary exponents appear, which correspond to different effective Luttinger liquid parameters. Even though the system consists of two separate interacting parts, the fixed point is described by a single conformally invariant boundary theory. We present details of the general effective bosonic field theory including the mode expansion and the finite size spectrum. The results are confirmed by numerical quantum Monte Carlo simulations on spinless fermions. We predict characteristic experimental signatures of the local density of states near junctions.

Scaling behavior and morphological properties of the interfaces obtained by the multilayer deposition process

The aim of the present work was to study numerically the scaling behavior and the morphological properties of the interfaces generated by the multilayer deposition process. We have noticed that, in the case where the ratio of the surface diffusion coefficient to the deposition rate reaches high values D/F>>1, the interface consists of mound structures. By using the dynamic scaling, we have shown that the height–height correlation function scales with time t and length l as G(l,t)~lαf(t/lα/β) with β=0.25±0.05 and α=0.51±0.02. These exponent values are equal to the ones predicted by the Edwards–Wilkinson approach. Besides, our results are in agreement with the growth system of Cu/Cu(100) at 300K which has been characterized in more detail by a combined scanning tunneling microscopy and spot profile analysis — low energy electronic diffusion study. Moreover, by considering two different methods, we have examined the fractal aspect of the obtained interfaces.

Advection of passive magnetic field by the Gaussian velocity field with finite correlations in time and spatial parity violation

Using the field theoretic renormalization group technique the model of passively advected weak magnetic field by an incompressible isotropic helical turbulent flow is investigated up to the second order of the perturbation theory (two-loop approximation) in the framework of an extended Kazantsev-Kraichnan model of kinematic magnetohydrodynamics. Statistical fluctuations of the velocity field are taken in the form of a Gaussian distribution with zero mean and defined noise with finite correlations in time. The two-loop analysis of all possible scaling regimes is done and the influence of helicity on the stability of scaling regimes is discussed and shown in the plane of exponents ɛ − η, where ɛ characterizes the energy spectrum of the velocity field in the inertial range E ∞ k 1 − 2e, and η is related to the correlation time at the wave number k which is scaled as k −2 + η. It is shown that in non-helical case the scaling regimes of the present vector model are completely identical and have also the same properties as those obtained in the corresponding model of passively advected scalar field. Besides, it is also shown that when the turbulent environment under consideration is helical then the properties of the scaling regimes in models of passively advected scalar and vector (magnetic) fields are essentially different. The results demonstrate the importance of the presence of a symmetry breaking in a given turbulent environment for investigation of the influence of an internal tensor structure of the advected field on the inertial range scaling properties of the model under consideration and will be used in the analysis of the influence of helicity on the anomalous scaling of correlation functions of passively advected magnetic field.

Scaling of correlation functions near capillary condensation

We study the influence of wetting films on (density–density) correlations for a fluid in a slit-like geometry near capillary condensation. We show that, for systems with short-ranged forces, the interaction between the wetting films strongly enhances the amplitude of the exponential decay of correlations and, unlike the interfacial roughness, is independent of a high-momentum cut-off. The correlation function shows scaling behaviour near condensation arising from the equality of two characteristic length scales: the parallel correlation length (associated with the complete wetting films) and a length scale related to the non-local interaction between the wetting films on either side of the slit. We introduce a dimensionless amplitude ratio associated with the decay of correlations which allows us to distinguish between local and non-local effective Hamiltonian theories. Only the latter is fully consistent with microscopic density functional descriptions of correlation functions. The influence of long-ranged intermolecular forces and fluctuation effects in two dimensions is also discussed.

Finite-size scaling of correlation functions in the one-dimensional Anderson-Hubbard model

We study the one-dimensional Anderson-Hubbard model using the density-matrix renormalization group method. The influence of disorder on the Tomonaga-Luttinger liquid behavior is quantitatively discussed. Based on the finite-size scaling analysis of density-density correlation functions, we find the following results: i) the charge exponent is significantly reduced by disorder at low filling and near half filling, ii) the localization length decays as $\xi \sim \Delta^{-2}$, where $\Delta$ is the disorder strength, independently of the on-site Coulomb interaction as well as band filling, and iii) the localization length is strongly suppressed by the on-site Coulomb interaction near half filling in association with the formation of the Mott plateaus.

Fractal physiology and the fractional calculus: a perspective.

This paper presents a restricted overview of Fractal Physiology focusing on the complexity of the human body and the characterization of that complexity through fractal measures and their dynamics, with fractal dynamics being described by the fractional calculus. Not only are anatomical structures (Grizzi and Chiriva-Internati, 2005), such as the convoluted surface of the brain, the lining of the bowel, neural networks and placenta, fractal, but the output of dynamical physiologic networks are fractal as well (Bassingthwaighte et al., 1994). The time series for the inter-beat intervals of the heart, inter-breath intervals and inter-stride intervals have all been shown to be fractal and/or multifractal statistical phenomena. Consequently, the fractal dimension turns out to be a significantly better indicator of organismic functions in health and disease than the traditional average measures, such as heart rate, breathing rate, and stride rate. The observation that human physiology is primarily fractal was first made in the 1980s, based on the analysis of a limited number of datasets. We review some of these phenomena herein by applying an allometric aggregation approach to the processing of physiologic time series. This straight forward method establishes the scaling behavior of complex physiologic networks and some dynamic models capable of generating such scaling are reviewed. These models include simple and fractional random walks, which describe how the scaling of correlation functions and probability densities are related to time series data. Subsequently, it is suggested that a proper methodology for describing the dynamics of fractal time series may well be the fractional calculus, either through the fractional Langevin equation or the fractional diffusion equation. A fractional operator (derivative or integral) acting on a fractal function, yields another fractal function, allowing us to construct a fractional Langevin equation to describe the evolution of a fractal statistical process. Control of physiologic complexity is one of the goals of medicine, in particular, understanding and controlling physiological networks in order to ensure their proper operation. We emphasize the difference between homeostatic and allometric control mechanisms. Homeostatic control has a negative feedback character, which is both local and rapid. Allometric control, on the other hand, is a relatively new concept that takes into account long-time memory, correlations that are inverse power law in time, as well as long-range interactions in complex phenomena as manifest by inverse power-law distributions in the network variable. We hypothesize that allometric control maintains the fractal character of erratic physiologic time series to enhance the robustness of physiological networks. Moreover, allometric control can often be described using the fractional calculus to capture the dynamics of complex physiologic networks.

Structure of correlation functions in single-field inflation

Many statistics available to constrain non-Gaussianity from inflation are simplest to use under the assumption that the curvature correlation functions are hierarchical. That is, if the n-point function is proportional to the (n-1) power of the two-point function amplitude and the fluctuations are small, the probability distribution can be approximated by expanding around a Gaussian in moments. However, single-field inflation with higher derivative interactions has a second small number, the sound speed, that appears in the problem when non-Gaussianity is significant and changes the scaling of correlation functions. Here we examine the structure of correlation functions in the most general single scalar field action with higher derivatives, formalizing the conditions under which the fluctuations can be expanded around a Gaussian distribution. We comment about the special case of the Dirac-Born-Infeld action.

Conformation of Circular DNA in Two Dimensions

The conformation of circular DNA molecules of various lengths adsorbed in a 2D conformation on a mica surface is studied. The results confirm the conjecture that the critical exponent nu is topologically invariant and equal to the self-avoiding walk value (in the present case nu=3/4), and that the topology and dimensionality of the system strongly influence the crossover between the rigid regime and the self-avoiding regime at a scale L approximately 7l{p}. Additionally, the bond correlation function scales with the molecular length L as predicted. For molecular lengths L<or=5l{p}, circular DNA behaves like a stiff molecule with an approximately elliptic shape.

Two-parameter scaling of correlation functions near continuous phase transitions

We discuss the order parameter correlation function in the vicinity of continuous phase transitions using a two-parameter scaling form G(k)=kc(-2)g(kxi,k/kc), where k is the wave vector and xi is the correlation length, and the interaction-dependent nonuniversal momentum scale kc remains finite at the critical fixed point. The correlation function describes the entire critical regime and captures the classical to critical crossover. One-parameter scaling is recovered only in the limit k/kc-->0. We present an approximate calculation of g(x,y) for the Ising universality class using the functional renormalization group.