Spontaneous magnetization of the Ising model on the Sierpinski carpet fractal, a rigorous result

We give a rigorous proof of the existence of spontaneous magnetization at finite temperature for the Ising spin model defined on the Sierpinski carpet fractal. The theorem is inspired by the classical Peierls argument for the two dimensional lattice. Therefore, this exact result proves the existence of spontaneous magnetization for the Ising model in low dimensional structures, i.e. structures with dimension smaller than 2.Comment: 14 pages, 8 figure

Multifractals of Normalized First Passage Time in Sierpinski Gasket

The multifractal behavior of the normalized first passage time is investigated on the two dimensional Sierpinski gasket with both absorbing and reflecting barriers. The normalized first passage time for Sinai model and the logistic model to arrive at the absorbing barrier after starting from an arbitrary site, especially obtained by the calculation via the Monte Carlo simulation, is discussed numerically. The generalized dimension and the spectrum are also estimated from the distribution of the normalized first passage time, and compared with the results on the finitely square lattice.Comment: 10 pages, Latex, with 3 figures and 1 table. to be published in J. Phys. Soc. Jpn. Vol.67(1998

On large deviation properties of Erdos-Renyi random graphs

We show that large deviation properties of Erd\"os-R\'enyi random graphs can be derived from the free energy of the $q$-state Potts model of statistical mechanics. More precisely the Legendre transform of the Potts free energy with respect to $\ln q$ is related to the component generating function of the graph ensemble. This generalizes the well-known mapping between typical properties of random graphs and the $q\to 1$ limit of the Potts free energy. For exponentially rare graphs we explicitly calculate the number of components, the size of the giant component, the degree distributions inside and outside the giant component, and the distribution of small component sizes. We also perform numerical simulations which are in very good agreement with our analytical work. Finally we demonstrate how the same results can be derived by studying the evolution of random graphs under the insertion of new vertices and edges, without recourse to the thermodynamics of the Potts model.Comment: 38 pages, 9 figures, Latex2e, corrected and extended version including numerical simulation result

Fractal chemical kinetics: Reacting random walkers

Computer simulations on binary reactions of random walkers ( A + A → A ) on fractal spaces bear out a recent conjecture: ( ρ −1 − ρ 0 −1 ) ∞ t f , where ρ is the instantaneous walker density and ρ 0 the initial one, and f = d s /2, where d s is the spectral dimension. For the Sierpinski gaskets: d =2, 2 f =1.38 ( d s =1.365); d =3, 2 f =1.56 ( d s =1.547); biased initial random distributions are compared to unbiased ones. For site percolation: d = 2, p =0.60, 2 f = 1.35 ( d s =1.35); d=3, p =0.32, 2 f =1.37 ( d s =1.4); fractal-to-Euclidean crossovers are also observed. For energetically disordered lattices, the effective 2 f (from reacting walkers) and d s (from single walkers) are in good agreement, in both two and three dimensions.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/45149/1/10955_2005_Article_BF01012924.pd

Single random walker on disordered lattices

Random walks on square lattice percolating clusters were followed for up to 2×105 steps. The mean number of distinct sites visited 〈 (S N ⊃> gives a spectral dimension of d s = 1.30±0.03 consistent with superuniversality ( d s =4J3) but closer to the alternative d s = 182/139, based on the low dimensionality correction. Simulations are also given for walkers on an energetically disordered lattice, with a jump probability that depends on the local energy mismatch and the temperature. An apparent fractal behavior is observed for a low enough reduced temperature. Above this temperature, the walker exhibits a “crossover” from fractal-to-Euclidean behavior. Walks on two- and three-dimensional lattices are similar, except that those in three dimensions are more efficient.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/45148/1/10955_2005_Article_BF01012923.pd