有限分枝分形上的自迴避迹行走

本文考虑在Sierpinski gasket及分支Koch曲线上的自迴避迹行走,运用实空间重整化群技术求出了相应的关联长度临界指数ν。结果表明,在Sierpinski gasket上,自迴避迹行走与自迴避行走属同一普适类;而在较高分枝度(R_max>3)的Koch曲线上,两者属不同普适类。 关键词：     

Thresholds and Universality of the Site Percolation on the Sierpinski Carpets

Following the methods proposed by Yonezawa, Sakamoto and Hori, we have calculated the percolation thresholds P_c, their error bars ΔP_c, and the correlation length exponents v of a family of the Sierpinski carpets for the site percolation problems by making use of MonteCarlo simulations and finite size scaling. We have found the dependence of P_c and v on the fractal dimensionality D_f and the lacunarity. We ascertain that the site percolation problems on a family of Sierpinski carpets with central cutouts and different D belong to different universal classes, and those on Sierpinski carpets with same D_f but of different lacunarities belong to different universal classes.     

Experimental study of the conductivity exponent for some Sierpinski carpets

The conductivity exponents in two dimensions for some Sierpinski carpets made of conductive paper are measured. The results show that the conductivity scaling behavior is highly dependent on symmetry for continuous fractal models. Compared with Sheng and Tao's theoretical calculation for elastic scaling behavior of a continuous Sierpinski carpet, the conductivity exponent is significantly less than the elasticity exponent. It might also show that the conductivity and elasticity belong to two different universality classes.     

Critical exponent for the loop erased self-avoiding walk by Monte Carlo methods

A Monte Carlo simulation was performed for loop-erased self-avoiding walks (LESAW) to ascertain the exponentv for the Z² and Z³ lattices. The estimated values were 2v=1.600±0.006 in two dimensions and 2v=1.232±0.008 in three dimensions, leading to the conjecturev=4/5 for the two-dimensional LESAW. These results add to existing evidence that the loop-erased self-avoiding walks are not in the same universality class as self-avoiding walks.     

Comment on "Critical behavior of the chain-generating function of self-avoiding walks on the sierpinski gasket family: the euclidean limit"

We refute the claims made by Riera and Chalub [Phys. Rev. E 58, 4001 (1998)] by demonstrating that they have not provided enough data (requisite in their series expansion method) to draw reliable conclusions about criticality of self-avoiding walks on the Sierpinski gasket family of fractals.     

Mean Field Theory of Ising Model on Sierpinski

We study Ising model on Sierpinski carpets by using mean field theory. We find a phase transition at T_c > 0 which is dependent on the geometrical factors. The critical exponents are calculated and found to be the same as the values for translationally invariant lattices.     

一簇金刚石晶格上S~4模型的相变

采用重整化群和累积展开的方法,研究了一簇金刚石晶格上S~4模型的相变,求得了系统的临界点.结果表明:当分支数m=2和m 12时,该系统只存在一个Gauss不动点K~*=b_2/2, u_2~*=0;当分支数3≤m≤12时,该系统不仅有Gauss不动点,还存在一个Wilson-Fisher不动点,并且后一个不动点对系统的临界特性产生决定性的影响.     

Adsorption of a flexible self-avoiding polymer chain: Exact results on fractal lattices

The large-scale behavior of surface-interacting self-avoiding polymer chains placed on finitely ramified fractal lattices is studied using exact recursion relations. It is shown how to obtain surface susceptibility critical indices and how to modify a scaling relation for these indices in the case of fractal lattices. We present the exact results for critical exponents at the point of adsorption transition for polymer chains situated on a class of Sierpinski gasket-type fractals. We provide numerical evidence for a critical behavior of the type found recently in the case of bulk self-avoiding random walks at the fractal to Euclidean crossover.     

Semi-flexible interacting self-avoiding trails on the square lattice

Self-avoiding walks self-interacting via nearest neighbours (ISAW) and self-avoiding trails interacting via multiply-visited sites (ISAT) are two models of the polymer collapse transition of a polymer in a dilute solution. On the square lattice it has been established numerically that the collapse transition of each model lies in a different universality class.     

Critical Percolation Probabilities for the Next-Nearest-Neighbouring Site Problems on Sierpinski Carpets

In this paper, we compute the next-nearest-neighbouring site percolation (Connections exist not only between nearest-neighbouring sites, but also between next-nearest-neigh bouring sites.) probabilities PC on the two-dimensional Sierpinski carpets, using the translationaldilation method and Monte Carlo technique. We obtain a relation among PC, fractal dimensionality D and connectivity Q. For the family of carpets with central cutouts,(1 - P_c)/(1 - P_c^s) = (D - 1)^1.60, where P_c^s = 0.41, the critical percolation probability for the next-nearest-neighbouring site problem on square lattice. As D reaches 2, P_c = P_c^s = 0.41, which is in agreement with the critical percolation probability on 2-d square lattices with . next-nearest-neigh bouring interactions.     

Critical Exponents of Fully Directed Flights: A Laplacian Transformation Method

A Laplacian transformation method is presented for d-dimensional fully directed flights (FDFs) with continuous steplengths. The correlation exponents we obtained are in agreement with the simulation study for fully directed Levy flight and the analytical results for FDF on a simple lattice. We believe that all FDFs in isotropic space belong to the same universality class.     

2+1维刻蚀模型生长表面等高线的共形不变性研究

为了更全面、有效地研究刻蚀模型(etching model)涨落表面的统计性质,基于Schramm Loewner Evolution(SLEκ)理论,对2+1维刻蚀模型饱和表面的等高线进行了数值模拟分析.研究表明,2+1维刻蚀模型饱和表面的等高线是共形不变曲线,可用Schramm Loewner Evolution理论进行描述,且扩散系数κ=2.70±0.04,属κ=8/3普适类.相应的等高线分形维数为df=1.34±0.01.     

On Continuous-Time Self-Avoiding Random Walk in Dimension Four

We study the distribution of the end-to-end distance of continuous-time self-avoiding random walks (CTRW) in dimension four from two viewpoints. From a real-space renormalization-group map on probabilities, we conjecture the asymptotic behavior of the end-to-end distance of a weakly self-avoiding random walk (SARW) that penalizes two-body interactions of random walks in dimension four on a hierarchical lattice. Then we perform the Monte Carlo computer simulations of CTRW on the four-dimensional integer lattice, paying special attention to the difference in statistical behavior of the CTRW compared with the discrete-time random walks. In this framework, we verify the result already predicted by the renormalization-group method and provide new results related to enumeration of self-avoiding random walks and calculation of the mean square end-to-end distance and gyration radius of continous-time self-avoiding random walks.     

Scaling of a collapsed polymer globule in two dimensions

Extensive Monte Carlo data analysis gives clear evidence that collapsed linear polymers in two dimensions fall in the universality class of athermal, dense self-avoiding walks, as conjectured by Duplantier [Phys. Rev. Lett. 71, 4274 (1993)].10.1103/PhysRevLett.71.4274 However, the boundary of the globule has self-affine roughness and does not determine the anticipated nonzero topological boundary contribution to entropic exponents. Scaling corrections are due to subleading contributions to the partition function corresponding to polymer configurations with one end located on the globule-solvent interface.     

Correction-to-Scaling Exponents for Two-Dimensional Self-Avoiding Walks

We study the correction-to-scaling exponents for the two-dimensional self-avoiding walk, using a combination of series-extrapolation and Monte Carlo methods. We enumerate all self-avoiding walks up to 59 steps on the square lattice, and up to 40 steps on the triangular lattice, measuring the mean-square end-to-end distance, the mean-square radius of gyration and the mean-square distance of a monomer from the endpoints. The complete endpoint distribution is also calculated for self-avoiding walks up to 32 steps (square) and up to 22 steps (triangular). We also generate self-avoiding walks on the square lattice by Monte Carlo, using the pivot algorithm, obtaining the mean-square radii to ≈ 0.01% accuracy up to N=4000. We give compelling evidence that the first non-analytic correction term for two-dimensional self-avoiding walks is Δ₁=3/2. We compute several moments of the endpoint distribution function, finding good agreement with the field-theoretic predictions. Finally, we study a particular invariant ratio that can be shown, by conformal-field-theory arguments, to vanish asymptotically, and we find the cancellation of the leading analytic correction.     

Angular structure of lacunarity, and the renormalization group

We formulate the angular structure of lacunarity in fractals, in terms of a symmetry reduction of the three point correlation function. This provides a rich probe of universality, and first measurements yield new evidence in support of the equivalence between self-avoiding walks (SAW's) and percolation perimeters in two dimensions. We argue that the lacunarity reveals much of the renormalization group in real space. This is supported by exact calculations for random walks and measured data for percolation clusters and SAW's. Relationships follow between exponents governing inward and outward propagating perturbations, and we also find a very general test for the contribution of long-range interactions.     

Statistical mechanics of random paths on disordered lattices

The dependence of the universality class on the statistical weight of unrestricted random paths is explicitly shown both for deterministic and statistical fractals such as the incipient infinite percolation cluster. Equally weighted paths (ideal chain) and kinetically generated paths (random walks) belong, in general, to different universality classes. For deterministic fractals exact renormalization group techniques are used. Asymptotic behaviors for the end-to-end distance ranging from power to logarithmic (localization) laws are observed for the ideal chain. In all these cases, random walks in the presence of nonperfect traps are shown to be in the same universality class of the ideal chain. Logarithmic behavior is reflected insingular renormalization group recursions. For the disordered case, numerical transfer matrix techniques are exploited on percolation clusters in two and three dimensions. The two-point correlation function scales with critical exponents not obeying standard scaling relations. The distribution of the number of chains and the number of chains returning to the starting point are found to be well approximated by a log-normal distribution. The logmoment of the number of chains is found to have an essential type of singularity consistent with the log-normal distribution. A non-self-averaging behavior is argued to occur on the basis of the results.     

Random walk statistics on fractal structures

We consider some statistical properties of simple random walks on fractal structures viewed as networks of sites and bonds: range, renewal theory, mean first passage time, etc. Asymptotic behaviors are shown to be controlled by the fractal (¯d) and spectral (¯d) dimensionalities of the considered structure. A simple decimation procedure giving the value of (¯d) is outlined and illustrated in the case of the Sierpinski gaskets. Recent results for the trapping problem, the self-avoiding walk, and the true-self-avoiding walk are briefly reviewed. New numerical results for diffusion on percolation clusters are also presented.     

Collision Statistics of Driven Polydisperse Granular Gases

We present a dynamical model of two-dimensional polydisperse granular gases with fractal size distribution, in which the disks are subject to inelastic mutual collisions and driven by standard white noise. The inhomogeneity of the disk size distribution can be measured by a fractal dimension df. By Monte Carlo simulations, we have mainly investigated the effect of the inhomogeneity on the statistical properties of the system in the same inelasticity case. Some novel results are found that the average energy of the system decays exponentially with a tendency to achieve a stable asymptotic value, and the system finally reaches a nonequilibrium steady state after a long evolution time. Furthermore, the inhomogeneity has great influence on the steady-state statistical properties. With the increase of the fractal dimension df, the distributions of path lengths and free times between collisions deviate more obviously from expected theoretical forms for elastic spheres and have an overpopulation of short distances and time bins. The collision rate increases with df, but it is independent of time. Meanwhile, the velocity distribution deviates more strongly from the Gaussian one, but does not demonstrate any apparent universal behavior.     

Simultaneous studies of the elasticity and the conductivity exponents of Sierpinski carpets

Simultaneous studies of the conductivity and the clasticity exponents on Sierpinski carpet made of metal and voids are reported. The elasticity exponent is =0.29±0.01, the conductivity exponent =0.22±0.01. It is obvious that they belong to different universality classes.