Low-density series expansions for directed percolation II: The square lattice with a wall

A new algorithm for the derivation of low-density expansions has been used to greatly extend the series for moments of the pair-connectedness on the directed square lattice near an impenetrable wall. Analysis of the series yields very accurate estimates for the critical point and exponents. In particular, the estimate for the exponent characterizing the average cluster length near the wall, $\tau_1=1.00014(2)$, appears to exclude the conjecture $\tau_1=1$. The critical point and the exponents $\nu_{\parallel}$ and $\nu_{\perp}$ have the same values as for the bulk problem.Comment: 8 pages, 1 figur

Nonuniversal Critical Spreading in Two Dimensions

Continuous phase transitions are studied in a two dimensional nonequilibrium model with an infinite number of absorbing configurations. Spreading from a localized source is characterized by nonuniversal critical exponents, which vary continuously with the density phi in the surrounding region. The exponent delta changes by more than an order of magnitude, and eta changes sign. The location of the critical point also depends on phi, which has important implications for scaling. As expected on the basis of universality, the static critical behavior belongs to the directed percolation class.Comment: 21 pages, REVTeX, figures available upon reques

Low-density series expansions for directed percolation III. Some two-dimensional lattices

We use very efficient algorithms to calculate low-density series for bond and site percolation on the directed triangular, honeycomb, kagom\'e, and $(4.8^2)$ lattices. Analysis of the series yields accurate estimates of the critical point $p_c$ and various critical exponents. The exponent estimates differ only in the $5^{th}$ digit, thus providing strong numerical evidence for the expected universality of the critical exponents for directed percolation problems. In addition we also study the non-physical singularities of the series.Comment: 20 pages, 8 figure

Osculating and neighbour-avoiding polygons on the square lattice

We study two simple modifications of self-avoiding polygons. Osculating polygons are a super-set in which we allow the perimeter of the polygon to touch at a vertex. Neighbour-avoiding polygons are only allowed to have nearest neighbour vertices provided these are joined by the associated edge and thus form a sub-set of self-avoiding polygons. We use the finite lattice method to count the number of osculating polygons and neighbour-avoiding polygons on the square lattice. We also calculate their radius of gyration and the first area-weighted moment. Analysis of the series confirms exact predictions for the critical exponents and the universality of various amplitude combinations. For both cases we have found exact solutions for the number of convex and almost-convex polygons.Comment: 14 pages, 5 figure

One Dimensional Nonequilibrium Kinetic Ising Models with Branching Annihilating Random Walk

Nonequilibrium kinetic Ising models evolving under the competing effect of spin flips at zero temperature and nearest neighbour spin exchanges at $T=\infty$ are investigated numerically from the point of view of a phase transition. Branching annihilating random walk of the ferromagnetic domain boundaries determines the steady state of the system for a range of parameters of the model. Critical exponents obtained by simulation are found to agree, within error, with those in Grassberger's cellular automata.Comment: 10 pages, Latex, figures upon request, SZFKI 05/9

Fear and its implications for stock markets

The value of stocks, indices and other assets, are examples of stochastic processes with unpredictable dynamics. In this paper, we discuss asymmetries in short term price movements that can not be associated with a long term positive trend. These empirical asymmetries predict that stock index drops are more common on a relatively short time scale than the corresponding raises. We present several empirical examples of such asymmetries. Furthermore, a simple model featuring occasional short periods of synchronized dropping prices for all stocks constituting the index is introduced with the aim of explaining these facts. The collective negative price movements are imagined triggered by external factors in our society, as well as internal to the economy, that create fear of the future among investors. This is parameterized by a ``fear factor'' defining the frequency of synchronized events. It is demonstrated that such a simple fear factor model can reproduce several empirical facts concerning index asymmetries. It is also pointed out that in its simplest form, the model has certain shortcomings.Comment: 5 pages, 5 figures. Submitted to the Proceedings of Applications of Physics in Financial Analysis 5, Turin 200

Perimeter Generating Functions For The Mean-Squared Radius Of Gyration Of Convex Polygons

We have derived long series expansions for the perimeter generating functions of the radius of gyration of various polygons with a convexity constraint. Using the series we numerically find simple (algebraic) exact solutions for the generating functions. In all cases the size exponent $\nu=1$.Comment: 8 pages, 1 figur

Critical behavior of a one-dimensional monomer-dimer reaction model with lateral interactions

A monomer-dimer reaction lattice model with lateral repulsion among the same species is studied using a mean-field analysis and Monte Carlo simulations. For weak repulsions, the model exhibits a first-order irreversible phase transition between two absorbing states saturated by each different species. Increasing the repulsion, a reactive stationary state appears in addition to the saturated states. The irreversible phase transitions from the reactive phase to any of the saturated states are continuous and belong to the directed percolation universality class. However, a different critical behavior is found at the point where the directed percolation phase boundaries meet. The values of the critical exponents calculated at the bicritical point are in good agreement with the exponents corresponding to the parity-conserving universality class. Since the adsorption-reaction processes does not lead to a non-trivial local parity-conserving dynamics, this result confirms that the twofold symmetry between absorbing states plays a relevant role in determining the universality class. The value of the exponent $\delta_2$, which characterizes the fluctuations of an interface at the bicritical point, supports the Bassler-Brown's conjecture which states that this is a new exponent in the parity-conserving universality class.Comment: 19 pages, 22 figures, to be published in Phys. Rev

Dimensional reduction in a model with infinitely many absorbing states

Using Monte Carlo method we study a two-dimensional model with infinitely many absorbing states. Our estimation of the critical exponent beta=0.273(5) suggests that the model belongs to the (1+1) rather than (2+1) directed-percolation universality class. We also show that for a large class of absorbing states the dynamic Monte Carlo method leads to spurious dynamical transitions.Comment: 6 pages, 4 figures, Phys.Rev. E, Dec. 199