Scaling in quantum gravity

The 2-point function is the natural object in quantum gravity for extracting critical behavior: The exponential falloff of the 2-point function with geodesic distance determines the fractal dimension d_H of space-time. The integral of the 2-point function determines the entropy exponent γ, i.e. the fractal structure related to baby universes, while the short distance behavior of the 2-point function connects γ and d_H by a quantum gravity version of Fisher's scaling relation. We verify this behavior in the case of 2d gravity by explicit calculation.     

Flocculation and percolation in reversible cluster-cluster aggregation

Off-lattice dynamic Monte-Carlo simulations were done of reversible cluster-cluster aggregation for spheres that form rigid bonds at contact. The equilibrium properties were found to be determined by the life time of encounters between two particles (t_e). t_e is a function not only of the probability to form or break a bond, but also of the elementary step size of the Brownian motion of the particles. In the flocculation regime the fractal dimension of the clusters is d_f=2.0 and the size distribution has a power law decay with exponent τ=1.5. At larger values of t_e transient gels are formed. Close to the percolation threshold the clusters have a fractal dimension d_f=2.7 and the power law exponent of the size distribution is τ=2.1. The transition between flocculation and percolation occurs at a characteristic weight average aggregation number that decreases with increasing volume fraction.     

Structural properties and scaling of the radius of gyration of two-dimensional star-branched polymers grown by diffusion

We built up star-branched polymers, whose morphology is fully determined by diffusion, with p=1,3,6 and 12 branches with a total of 30,000 monomer units. We investigated their structural properties by calculating the monomer-monomer correlation functions. A detailed finite size scaling analysis of the radius of gyration was also performed to determine the exponent and the corrections to scaling. From these results we calculated the fractal dimension of the branched aggregates and obtained: d_f=1.222(7), for the linear chain, d_f=1.2305(8), d_f=1.247(8) and d_f=1.261(8) for the three, six and twelve branches polymer, respectively.     

Interplay of Quantum and Classical Fluctuations Near Quantum Critical Points

For a system near a quantum critical point (QCP), above its lower critical dimension d _L , there is in general a critical line of second-order phase transitions that separates the broken symmetry phase at finite temperatures from the disordered phase. The phase transitions along this line are governed by thermal critical exponents that are different from those associated with the quantum critical point. We point out that, if the effective dimension of the QCP, d _eff?=?d?+?z (d is the Euclidean dimension of the system and z the dynamic quantum critical exponent) is above its upper critical dimension $d_{_{C}}$ there is an intermingle of classical (thermal) and quantum critical fluctuations near the QCP. This is due to the breakdown of the generalized scaling relation ψ?=?νz between the shift exponent ψ of the critical line and the crossover exponent νz, for $d+z>d_{_{C}}$ by a dangerous irrelevant interaction. This phenomenon has clear experimental consequences, like the suppression of the amplitude of classical critical fluctuations near the line of finite temperature phase transitions as the critical temperature is reduced approaching the QCP.     

Circular-like maps: sensitivity to the initial conditions, multifractality and nonextensivity

Dissipative one-dimensional maps may exhibit special points (e.g., chaos threshold) at which the Lyapunov exponent vanishes. Consistently, the sensitivity to the initial conditions has a power-law time dependence, instead of the usual exponential one. The associated exponent can be identified with 1/(1-q), where q characterizes the nonextensivity of a generalized entropic form currently used to extend standard, Boltzmann-Gibbs statistical mechanics in order to cover a variety of anomalous situations. It has been recently proposed (Lyra and Tsallis, Phys. Rev. Lett. 80, 53 (1998)) for such maps the scaling law , where and are the extreme values appearing in the multifractal function. We generalize herein the usual circular map by considering inflexions of arbitrary power z, and verify that the scaling law holds for a large range of z. Since, for this family of maps, the Hausdorff dimension d_f equals unity for all z in contrast with q which does depend on z, it becomes clear that d_f plays no major role in the sensitivity to the initial conditions. Received 5 February 1999     

Critical behavior near the paramagnetic to ferromagnetic phase transition temperature in La0.7Pb0.05Na0.25MnO3

Critical behavior in La_0.7Pb_0.05Na_0.25MnO₃ has been investigated by dc magnetization measurements. Magnetic data indicate that the compound exhibits a continuous (second-order) paramagnetic (PM) to ferromagnetic (FM) phase transition. Estimates of critical exponents yield δ=4.80±0.01, γ=1.296±0.002 and β=0.344±0.007 (consistent with both the predictions for the three-dimensional-Heisenberg model and with those reported for materials when the FM transition is ascribed to the double exchange (DE) mechanism as a major origin) with T_C=334.54±0.08. The critical exponent γ is slightly inferior than predicted from the 3D Heisenberg model. Such a difference may be due, within the context of the quenched disorder, to the presence of some alterations of short-range magnetic order of FM clusters in the PM phase. The temperature variation in the effective exponent (γ_eff) is similar to those for disordered ferromagnets.     

On the upper critical dimension of Bernoulli percolation

We derive a set of inequalities for thed-dimensional independent percolation problem. Assuming the existence of critical exponents, these inequalities imply: $$\begin{gathered} f + v \geqq 1 + \beta _Q , \hfill \\ \mu + v \geqq 1 + \beta _Q , \hfill \\ \zeta \geqq \min \left\{ {1,\frac{{v^, }}{v}} \right\}, \hfill \\ \end{gathered} $$ where the above exponents aref: the flow constant exponent, ν(ν′): the correlation length exponent below (above) threshold, μ: the surface tension exponent, β _Q : the backbone density exponent and ζ: the chemical distance exponent. Note that all of these inequalities are mean-field bounds, and that they relate the exponentv defined from below the percolation threshold to exponents defined from above threshold. Furthermore, we combine the strategy of the proofs of these inequalities with notions of finite-size scaling to derive: $$\max \{ dv,dv^, \} \geqq 1 + \beta _Q ,$$ whered is the lattice dimension. Since β _Q ≧2β, where β is the percolation density exponent, the final bound implies that, below six dimensions, the standard order parameter and correlation length exponents cannot simultaneously assume their mean-field values; hence an implicit bound on the upper critical dimension:d _c ≧6.     

Spatial scaling in fracture propagation in dilute systems

The geometry of fracture patterns in a dilute elastic network is explored using molecular dynamics simulation. The network in two dimensions is subjected to a uniform strain which drives the fracture to develop by the growth and coalescence of the vacancy clusters in the network. For strong dilution, it has been shown earlier that there exists a characteristic time t_c at which a dynamical transition occurs with a power law divergence (with the exponent z) of the average cluster size. Close to t_c, the growth of the clusters is scale-invariant in time and satisfies a dynamical scaling law. This paper shows that the cluster growth near t_c also exhibits spatial scaling in addition to the temporal scaling. As fracture develops with time, the connectivity length xi of the clusters increases and diverges at t_c as xi ∼ (t_c − t)^−ν, with ν = 0.83 ± 0.06. As a result of the scale-invariant growth, the vacancy clusters attain a fractal structure at t_c with an effective dimensionality d_f ∼ 1.85 ± 0.05. These values are independent (within the limit of statistical error) of the concentration (provided it is sufficiently high) with which the network is diluted to begin with. Moreover, the values are very different from the corresponding values in qualitatively similar phenomena suggesting a different universality class of the problem. The values of ν and d_f supports the scaling relation z = νd_f with the value of z obtained before.     

Dynamic scaling behaviors of the restricted-solid-on-solid model on honeycomb and square-octagon lattice substrates

The dynamic scaling behaviors of the restricted-solid-on-solid (RSOS) model on two new types of substrate, which are honeycomb and square-octagon lattice substrates, are studied by means of Kinetic Monte Carlo simulations. The growth exponent β and the roughness exponent α defined, respectively, by the surface width via W ~ t ^β and the saturated width via W _sat ~ L ^α , L being the system size, were obtained by a power-counting analysis. Our simulation results show that the Family-Vicsek scaling is still satisfied. However, the structures of the substrates indeed affect the dynamic behavior of the growth model. The values of the roughness exponents fall between regular and fractal lattices. Deeper analysis show that the coordination number of the substrates play an crucial role.     

Majority-vote on directed Erd?s-Rényi random graphs

Through Monte Carlo Simulation, the well-known majority-vote model has been studied with noise on directed random graphs. In order to characterize completely the observed order-disorder phase transition, the critical noise parameter q_c, as well as the critical exponents β/ν, γ/ν and 1/ν have been calculated as a function of the connectivity z of the random graph.     

Height-height correlations for surface growth on percolation networks

The height-height correlations of the surface growth for equilibrium and nonequilibrium restricted solid-on-solid (RSOS) model were investigated on randomly diluted lattices, i.e., on infinite percolation networks. It was found that the correlation function calculated over the chemical distances reflected the dynamics better than that calculated over the geometrical distances. For the equilibrium growth on a critical percolation network, the correlation function for the evolution time t?1 yielded a power-law behavior with the power ζ^′, associated with the roughness exponent ζ via the relation ζ=ζ^′d_f/d_l, with d_f and d_l being, respectively, the fractal dimension and the chemical dimension of the substrate. For the nonequilibrium growth, on the other hand, the correlation functions did not yield power-law behaviors for the concentration of diluted sites x less than or equal to the critical concentration x_c.     

Vehicular traffic through a self-similar sequence of traffic lights

We study the dynamical behavior of vehicular traffic through a sequence of traffic lights positioned self-similarly on a highway, where all traffic lights turn on and off simultaneously with cycle time T_s. The signals are positioned self-similarly by Cantor set. The nonlinear-map model of vehicular traffic controlled by self-similar signals is presented. The vehicle exhibits the complex behavior with varying cycle time. The tour time is much lower such that signals are positioned periodically with the same interval. The arrival time T(x) at position x scales as (T(x)-x)∝x^d_f, where d_f is the fractal dimension of Cantor set. The landscape in the plot of T(x)−x against cycle time T_s shows a self-affine fractal with roughness exponent α=1−d_f.     

Mean-field and high temperature series expansion calculations of some magnetic properties of Ising and XY antiferromagnetic thin-films

In this work,the magnetic properties of Ising and XY antiferromagnetic thin-films are investigated each as a function of N’eel temperature and thickness for layers(n = 2,3,4,5,6,and bulk(∞)) by means of a mean-field and high temperature series expansion(HTSE) combined with Pad’e approximant calculations.The scaling law of magnetic susceptibility and magnetization is used to determine the critical exponent γ,ν eff(mean),ratio of the critical exponents γ/ν,and magnetic properties of Ising and XY antiferromagnetic thin-films for different thickness layers n = 2,3,4,5,6,and bulk(∞).     

Critical behavior of the 3D Ising model on a poissonian random lattice

The single-cluster Monte Carlo algorithm and the reweighting technique are used to simulate the 3D ferromagnetic Ising model on 3D Voronoi-Delauney lattices. It is assumed that the coupling factor J varies with the distance r between the first neighbors as J(r)∝e^−ar, with a≥0. The critical exponents γ/ν, β/ν, and ν are calculated, and according to the present estimates for the critical exponents, we argue that this random system belongs to the same universality class of the pure 3D ferromagnetic Ising model.     

Critical behavior studies in ferromagnetic (Nd, K)-Mn-O compounds

The critical scaling behavior of K-doped Nd-Mn-O based double-exchange ferromagnetic compounds was studied by measuring isothermal magnetization of Nd_0.84K_0.16MnO₃ and Nd_0.77K_0.23MnO₃ samples. The critical exponents β, γ and δ corresponding to the spontaneous magnetization, initial susceptibility and isothermal magnetization, respectively, were determined by analyzing the magnetization data in terms of the modified Arrott plot method. The critical exponent values of both samples are found to be comparable to values predicted by a mean field model. The role of ferromagnetic clusters on the scaling behavior is discussed. The critical exponent values are found to be consistent with the Widom scaling relation and the universal scaling hypothesis.     

Magnetic interaction and the critical dynamics above the curie point

It is shown that near T_c, where 4πχ ? 1 magnetic interaction completely change the long-wave critical dynamics of the ferromagnets and the new dynamic critical exponent is z_d = (5 - η)/2 - ν^-1 ≈ 1.     

Characterization of electrodeposited nickel film surfaces using atomic force microscopy

Microstructures of nickel surfaces electrodeposited on indium tin oxides coated glasses are investigated using atomic force microscopy. The fractal dimension D and Hurst exponent H of the nickel surface images are determined from a frequency analysis method proposed by Aguilar et al. [J. Microsc. 172 (1993) 233] and from Hurst rescaled range analysis. The two methods are found to give the same value of the fractal dimension D∼2.0. The roughness exponent α and growth exponent β that characterize scaling behaviors of the surface growth in electrodeposition are calculated using the height-difference correlation function and interface width in Fourier space. The exponents of α∼1.0 and β∼0.8 show that the surface growth does not belong to the universality classes theoretically predicted by statistical growth models.     

Precise measurement of Γ(K→e ν(γ))/Γ(K→μ ν(γ)) and study of K→e ν γ

We present a precise measurement of the ratio R _K =Γ(K→e ν(γ))/Γ(K→μ ν(γ)) and a study of the radiative process K→e ν γ, performed with the KLOE detector. The results are based on data collected at the Frascati e ⁺ e ^? collider DAΦNE for an integrated luminosity of 2.2 fb^?1. We find R _K =(2.493±0.025_stat±0.019_syst)×10^?5, in agreement with the Standard Model expectation. This result is used to improve constraints on parameters of the Minimal Supersymmetric Standard Model with lepton flavor violation. We also measured the differential decay rate dΓ(K→e ν γ)/dE _γ for photon energies 10<E _γ < 250 MeV. Results are compared with predictions from theory.     

Monte Carlo analysis of critical properties of the two-dimensional randomly site-diluted Ising model via Wang-Landau algorithm

The influence of random site dilution on the critical properties of the two-dimensional Ising model on a square lattice was explored by Monte Carlo simulations with the Wang-Landau sampling. The lattice linear size was L=20-120 and the concentration of diluted sites q=0.1,0.2,0.3. Its pure version displays a second-order phase transition with a vanishing specific heat critical exponent α, thus, the Harris criterion is inconclusive, in that disorder is a relevant or irrelevant perturbation for the critical behaviour of the pure system. The main effort was focused on the specific heat and magnetic susceptibility. We have also looked at the probability distribution of susceptibility, pseudocritical temperatures and specific heat for assessing self-averaging. The study was carried out in appropriate restricted but dominant energy subspaces. By applying the finite-size scaling analysis, the correlation length exponent ν was found to be greater than one, whereas the ratio of the critical exponents (α/ν) is negative and (γ/ν) retains its pure Ising model value supporting weak universality.