Two-dimensional Sierpinski Gasket Research Articles

Asymptotic behavior of a generalized independent sets model on the two-dimensional Sierpinski gasket

We consider a generalized independent set model on the two-dimensional Sierpinski gasket SGn. Let A be a set of symbols and each vertex of SGn should take a symbol from A. Given a subset S⊂A such that the symbols of nearest-neighbor vertices are not allowed to be the forbidden blocks F={ij:i,j∈S}. The probability of each vertex of SGn to take a symbol in S is denoted by p ∈ (0, 1), and the probability of each vertex to take a symbol not in S is given by 1 − p. When A={0,1}, S={0}, and p = 1/2, this reduces to independent set model or golden mean shift in symbolic dynamics. We investigate the asymptotic behavior of this generalized model on the two-dimensional Sierpinski gasket and obtain upper and lower bounds of the entropy per site for p ∈ (0, 1).

The ice model and the eight-vertex model on the two-dimensional Sierpinski gasket

We present the numbers of ice model and eight-vertex model configurations (with Boltzmann factors equal to one), I(n) and E(n) respectively, on the two-dimensional Sierpinski gasket SG(n) at stage $n$. For the eight-vertex model, the number of configurations is $E(n)=2^{3(3^n+1)/2}$ and the entropy per site, defined as $\lim_{v \to \infty} \ln E(n)/v$ where $v$ is the number of vertices on SG(n), is exactly equal to $\ln 2$. For the ice model, the upper and lower bounds for the entropy per site $\lim_{v \to \infty} \ln I(n)/v$ are derived in terms of the results at a certain stage. As the difference between these bounds converges quickly to zero as the calculated stage increases, the numerical value of the entropy can be evaluated with more than a hundred significant figures accurate. The corresponding result of ice model on the generalized two-dimensional Sierpinski gasket SG_b(n) with $b=3$ is also obtained. For the generalized vertex model on SG_3(n), the number of configurations is $2^{(8 \times 6^n +7)/5}$ and the entropy per site is equal to $\frac87 \ln 2$. The general upper and lower bounds for the entropy per site for arbitrary $b$ are conjectured.

ACYCLIC ORIENTATIONS ON THE SIERPINSKI GASKET

We study the number of acyclic orientations on the generalized two-dimensional Sierpinski gasket SG 2,b(n) at stage n with b equal to two and three, and determine the asymptotic behaviors. We also derive upper bounds for the asymptotic growth constants of SG 2,b and d-dimensional Sierpinski gasket SG d.

Hamiltonian walks on the Sierpinski gasket

We derive the exact number of Hamiltonian walks H(n) on the two-dimensional Sierpinski gasket SG(n) at stage n, whose asymptotic behavior is given by $\frac{\sqrt{3}(2\sqrt{3})^{3^{n-1}}}{3} \times (\frac{5^2 \times 7^2 \times 17^2}{2^{12} \times 3^5 \times 13})(16)^n$3(23)3n−13×(52×72×172212×35×13)(16)n. We also obtain the number of Hamiltonian walks with one end at a specific outmost vertex of SG(n), with asymptotic behavior $\frac{\sqrt{3}(2\sqrt{3})^{3^{n-1}}}{3} \times (\frac{7 \times 17}{2^4 \times 3^3})4^n$3(23)3n−13×(7×1724×33)4n. The distribution of Hamiltonian walks on SG(n) with one end at a specific outmost vertex and the other at an arbitrary vertex of SG(n) is investigated. We rigorously prove that the exponent for the mean ℓ displacement between the two end vertices of such Hamiltonian walks on SG(n) is ℓln 2/ln 3 for ℓ > 0.

Structure of spanning trees on the two-dimensional Sierpinski gasket

Combinatorics Consider spanning trees on the two-dimensional Sierpinski gasket SG(n) where stage n is a non-negative integer. For any given vertex x of SG(n), we derive rigorously the probability distribution of the degree j ∈{1,2,3,4} at the vertex and its value in the infinite n limit. Adding up such probabilities of all the vertices divided by the number of vertices, we obtain the average probability distribution of the degree j. The corresponding limiting distribution φj gives the average probability that a vertex is connected by 1, 2, 3 or 4 bond(s) among all the spanning tree configurations. They are rational numbers given as φ1=10957/40464, φ2=6626035/13636368, φ3=2943139/13636368, φ4=124895/4545456.

MULTIFRACTAL MEASURES CHARACTERIZED BY THE ITERATIVE MAP WITH TWO CONTROL PARAMETERS

A one-dimensional iterative map with two control parameters — the Kim–Kong map — is proposed. Our purpose is to investigate the characteristic properties of this map, and to discuss numerically the multifractal behavior of the normalized first passage time. Based especially on the Monte Carlo simulation, the normalized first passage time to arrive at the absorbing barrier after starting from an arbitrary site is mainly obtained in the presence of both absorption and reflection on a two-dimensional Sierpinski gasket. We also discuss the multifractal spectra of the normalized first passage time, and the numerical result of the Kim–Kong model presented will be compared with that of the Sinai and logistic models.

Dirichlet Forms and Brownian Motion Penetrating Fractals

Can a Brownian motion penetrate the two-dimensional Sierpinski gasket? This question was studied in [8], and an affirmative answer was given. In this paper, the problem is studied with a different approach, using Dirichlet forms and function space theory. The results obtained are somewhat different from, and from certain aspects more general than, the results in [8].

Sandpile model on the Sierpinski gasket fractal.

We investigate the sandpile model on the two-dimensional Sierpinski gasket fractal. We find that the model displays interesting critical behavior, and we analyze the distribution functions of avalanche sizes, lifetimes, and topplings and calculate the associated critical exponents t51.5160.04, a51.6360.04, and m51.3660.04. The avalanche size distribution shows power-law behavior modulated by logarithmic oscillations which can be related to the discrete scale invariance of the underlying lattice. Such a distribution can be formally described by introducing a complex scaling exponent t*[t1id, where the real part t corresponds to the power law and the imaginary part d is related to the period of the logarithmic oscillations. @S1063-651X~96!03907-4#

Level statistics for electronic states in a disordered fractal

We present results for the density of states and the statistics of the energy levels in a random tight binding matrix ensemble defined on a disordered two-dimensional Sierpinski gasket. In the absence of disorder the nearest level spacing distribution function P(S) is shown to follow the inverse power law , which defines the fractal dimension of the corresponding spectrum. In the random case P(S) approaches, instead, the Poisson law , which is consistent with localization of the corresponding eigenstates. In the presence of a random magnetic flux our results also scale towards the Poisson statistics.

Exact ground-state results for a disordered-continuum model of the halogen-bridged metal complex PtI

By mapping a continuum model of a halogen-bridged metal complex chain to the Takayama-Lin-Liu-Maki continuum model of polyacetylene and using a supersymmetric functional integral formalism, the authors study the influence of both 'site' and 'bond' disorder on the nature of the ground state of this halogen-bridged metal complex PtI. They find exact results for the amplitude of the order parameter Delta and the electronic density of states as a function of the strength of the disorder. They also obtain a phase diagram of Delta versus both bond and site disorder. The critical concentration of site impurities versus average impurity scattering strength gives a fractal dimension which is very close to that of the two-dimensional Sierpinski gasket.

Asymptotic form of the spectral dimension of the Sierpinski gasket type of fractals

The authors have studied the spectral dimension d48T of an infinite class of fractals. The first member (b=2) of the class is the two-dimensional Sierpinski gasket, while the last member (b= infinity ) appears to be a wedge of the ordinary triangular lattice. By studying the electric resistance of the fractals they have been able to calculate exact values of d for the first 200 members of the class. An analysis of the obtained data reveals that for large b the spectral dimension should approach the upper limit of 2 according to the formula d approximately=2-constant (ln b)beta , where beta is not larger than one. This result implies, among other things, that the scaling exponents of the resistivity and diffusion constant should logarithmically vanish at the fractal-lattice crossover.

Monte Carlo study of random walks on a 2D gasket fractal in an external field

Using Monte Carlo simulation, random walks on a two-dimensional Sierpinski gasket in the presence of an external field were studied. It was observed that the random walk motion of a particle displayed a crossover from anomalous diffusion to drift for a non-zero bias field, and the crossover time tcr was a decreasing function of the external bias field. The associated dynamic exponents obtained in a computer simulation agree with the predictions of Stinchcombe's scaling treatment (1985).

Critical exponents of the self-avoiding walks on a family of finitely ramified fractals

The authors have studied the self-avoiding walks (SAW) on a family of finitely ramified fractals. The first member (b=2) of the family is the two-dimensional Sierpinski gasket, while the last member (b= infinity ) appears to be a wedge of the homogeneous triangular lattice. By means of the exact renormalisation group transformations they have calculated the critical exponents alpha , nu and gamma , and the connectivity constant mu , of SAW on each member of a sequence (2<or=b<or=8) of the studied fractal family. The obtained exact results are compared with the recent phenomenological proposals and with the results believed to be exact in the case of a homogeneous two-dimensional lattice.

Critical dynamics of the kinetic Ising model on fractal geometries.

The critical dynamics of the kinetic Glauber-Ising model on different fractal geometries is studied. The classes of fractals which are examined are the nonbranching Koch curves, the branching Koch curves, and the two-dimensional Sierpinski gasket. The critical dynamic exponent is calculated for these models using an exact renormalization-group transformation. The value z=2.58 for the two-dimensional Sierpinski gasket agrees with recent results from experiments performed in a percolating system.

Vibrational spectrum on an exact fractal lattice

Abstract The vibrational properties were studied on the two-dimensional Sierpinski gasket. If one allows the mass on each site to move in two orthogonal directions, one needs to introduce a transverse coupling K in addition to the longitudinal couplings K1 and K2. Using the standard decimation procedures, one derives the rescaling properties of the couplings K1, K2 and K. One finds, if K is the geometric mean of K1, K2, then upon rescaling, the new coupling K' remains as the geometric mean of K'1 and K'2. By using the lattice Green's function and its rescaling properties, one is able to obtain the density of vibrational states as well as the correlation between two given sites. According to the correlation function, one finds there exists an energy dependent localization length ξ, such that when | r 1 − r 2 |>ξ, the ratios K/K1 and K/K2 go to constants upon rescaling. Also, in the limit of K = 0, the two orthogonal motions are completely decoupled and our results reduce to the case of one degree of freedom.