Self-avoiding walk on a square lattice with correlated vacancies

The self-avoiding walk on the square site-diluted correlated percolation lattice is considered. The Ising model is employed to realize the spatial correlations of the metric space. As a well-accepted result, the (generalized) Flory's mean-field relation is tested to measure the effect of correlation. After exploring a perturbative Fokker-Planck-like equation, we apply an enriched Rosenbluth Monte Carlo method to study the problem. To be more precise, the winding angle analysis is also performed from which the diffusivity parameter of Schramm-Loewner evolution theory (κ) is extracted. We find that at the critical Ising (host) system, the exponents are in agreement with Flory's approximation. For the off-critical Ising system, we find also a behavior for the fractal dimension of the walker trace in terms of the correlation length of the Ising system ξ(T), i.e., D_{F}^{SAW}(T)-D_{F}^{SAW}(T_{c})∼1/sqrt[ξ(T)].

Density of zeros of the ferromagnetic Ising model on a family of fractals

We studied distribution of zeros of the partition function of the ferromagnetic Ising model near the Yang-Lee edge on a family of Sierpinski gasket lattices whose members are labeled by an integer b (2 ≤ b<∞). The obtained exact results on the first seven members of this family show that, for b ≥ 4, associated correlation length diverges more slowly than any power law when distance δh from the edge tends to zero, ξ_{YL}∼exp[ln(b)sqrt[|ln(δh)|/ln(λ{0})]], λ{0} being a decreasing function of b. We suggest a possible scenario for the emergence of the usual power-law behavior in the limit of very large b when fractal lattices become almost compact.

Stiffness dependence of critical exponents of semiflexible polymer chains situated on two-dimensional compact fractals

We present an exact and Monte Carlo renormalization group (MCRG) study of semiflexible polymer chains on an infinite family of the plane-filling (PF) fractals. The fractals are compact, that is, their fractal dimension df is equal to 2 for all members of the fractal family enumerated by the odd integer b(3<or=b<infinity). For various values of stiffness parameter s of the chain, on the PF fractals (for 3<or=b<or=9 ), we calculate exactly the critical exponents nu (associated with the mean squared end-to-end distances of polymer chain) and gamma (associated with the total number of different polymer chains). In addition, we calculate nu and gamma through the MCRG approach for b up to 201. Our results show that for each particular b, critical exponents are stiffness dependent functions, in such a way that the stiffer polymer chains (with smaller values of s) display enlarged values of nu, and diminished values of gamma. On the other hand, for any specific s, the critical exponent nu monotonically decreases, whereas the critical exponent gamma monotonically increases, with the scaling parameter b. We reflect on a possible relevance of the criticality of semiflexible polymer chains on the PF family of fractals to the same problem on the regular Euclidean lattices.

Multifractality of self-avoiding walks on percolation clusters

We consider self-avoiding walks on the backbone of percolation clusters in space dimensions d=2,3,4. Applying numerical simulations, we show that the whole multifractal spectrum of singularities emerges in exploring the peculiarities of the model. We obtain estimates for the set of critical exponents that govern scaling laws of higher moments of the distribution of percolation cluster sites visited by self-avoiding walks, in a good correspondence with an appropriately summed field-theoretical epsilon=6-d expansion [H.-K. Janssen and O. Stenull, Phys. Rev. E 75, 020801(R) (2007)10.1103/PhysRevE.75.020801].

Scaling of Hamiltonian walks on fractal lattices

We investigate asymptotical behavior of numbers of long Hamiltonian walks (HWs), i.e., self-avoiding random walks that visit every site of a lattice, on various fractal lattices. By applying an exact recursive technique we obtain scaling forms for open HWs on three-simplex lattice, Sierpinski gasket, and their generalizations: Given-Mandelbrot (GM), modified Sierpinski gasket (MSG), and n -simplex fractal families. For GM, MSG and n -simplex lattices with odd values of n , the number of open HWs Z(N), for the lattice with N>>1 sites, varies as omega(N)}N(gamma). We explicitly calculate the exponent gamma for several members of GM and MSG families, as well as for n-simplices with n=3, 5, and 7. For n-simplex fractals with even n we find different scaling form: Z(N) approximately omega(N)mu(N1/d(f), where d(f) is the fractal dimension of the lattice, which also differs from the formula expected for homogeneous lattices. We discuss possible implications of our results on studies of real compact polymers.

Branched polymers on the Given-Mandelbrot family of fractals

We study the average number A (n) per site of the number of different configurations of a branched polymer of n bonds on the Given-Mandelbrot family of fractals using exact real-space renormalization. Different members of the family are characterized by an integer parameter b , 2 < or = b < or = infinity . The fractal dimension varies from log(2) 3 to 2 as b is varied from 2 to infinity. We find that for all b > or = 3 , A (n) varies as lambda(n) exp (b n(psi)) where lambda and b are some constants, and 0 < psi < 1 . We determine the exponent psi, and the size exponent nu (average diameter of polymer varies as n(nu) ), exactly for all b , 3 < or = b < o r= infinity . This generalizes the earlier results of Knezevic and Vannimenus for b = 3 [Phys. Rev B 35, 4988 (1987)].

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.

Dynamics of brain electrical activity

In addition to providing important theoretical insights into chaotic deterministic systems, dynamical systems theory has provided techniques for analyzing experimental data. These methods have been applied to a variety of physical and chemical systems. More recently, biological applications have become important. In this paper, we report applications of one of these techniques, estimation of a signal's correlation dimension, to the characterization of human electroencephalographic (EEG) signals and event-related brain potentials (ERPs). These calculations demonstrate that the magnitude of the technical difficulties encountered when attempting to estimate dimensions from noisy biological signals are substantial. However, these results also suggest that this procedure can provide a partial characterization of changes in cerebral electrical activity associated with changes in cognitive behavior that complements classical analytic procedures.