Stochastic Model and Equivalent Ferromagnetic Spin Chain with Alternation

We investigate a non-equilibrium reaction-diffusion model and equivalent ferromagnetic spin 1/2 XY spin chain with alternating coupling constant. The exact energy spectrum and the n-point hole correlations are considered with the help of the Jordan-Wigner fermionization and the inter-particle distribution function method. Although the Hamiltonian has no explicit translational symmetry, the translational invariance is recovered after long time due to the diffusion. We see the scaling relations for the concentration and the two-point function in finite size analysis.Comment: 7 pages, LaTeX file, to appear in J. Phys. A: Math. and Ge

Space Representation of Stochastic Processes with Delay

We show that a time series $x_t$ evolving by a non-local update rule $x_t = f (x_{t-n},x_{t-k})$ with two different delays $k<n$ can be mapped onto a local process in two dimensions with special time-delayed boundary conditions provided that $n$ and $k$ are coprime. For certain stochastic update rules exhibiting a non-equilibrium phase transition this mapping implies that the critical behavior does not depend on the short delay $k$. In these cases, the autocorrelation function of the time series is related to the critical properties of directed percolation.Comment: 6 pages, 8 figure

Weakly disordered absorbing-state phase transitions

The effects of quenched disorder on nonequilibrium phase transitions in the directed percolation universality class are revisited. Using a strong-disorder energy-space renormalization group, it is shown that for any amount of disorder the critical behavior is controlled by an infinite-randomness fixed point in the universality class of the random transverse-field Ising models. The experimental relevance of our results are discussed.Comment: 4 pages, 2 eps figures; (v2) references and discussion on experiments added; (v3) published version, minor typos corrected, some side discussions dropped due to size constrain

Matrix Product Eigenstates for One-Dimensional Stochastic Models and Quantum Spin Chains

We show that all zero energy eigenstates of an arbitrary $m$--state quantum spin chain Hamiltonian with nearest neighbor interaction in the bulk and single site boundary terms, which can also describe the dynamics of stochastic models, can be written as matrix product states. This means that the weights in these states can be expressed as expectation values in a Fock representation of an algebra generated by $2m$ operators fulfilling $m^2$ quadratic relations which are defined by the Hamiltonian.Comment: 11 pages, Late

On Matrix Product Ground States for Reaction-Diffusion Models

We discuss a new mechanism leading to a matrix product form for the stationary state of one-dimensional stochastic models. The corresponding algebra is quadratic and involves four different matrices. For the example of a coagulation-decoagulation model explicit four-dimensional representations are given and exact expressions for various physical quantities are recovered. We also find the general structure of $n$-point correlation functions at the phase transition.Comment: LaTeX source, 7 pages, no figure

A precise approximation for directed percolation in d=1+1

We introduce an approximation specific to a continuous model for directed percolation, which is strictly equivalent to 1+1 dimensional directed bond percolation. We find that the critical exponent associated to the order parameter (percolation probability) is beta=(1-1/\sqrt{5})/2=0.276393202..., in remarkable agreement with the best current numerical estimate beta=0.276486(8).Comment: 4 pages, 3 EPS figures; Submitted to Physical Review Letters v2: minor typos + 1 major typo in Eq. (30) correcte

From multiplicative noise to directed percolation in wetting transitions

A simple one-dimensional microscopic model of the depinning transition of an interface from an attractive hard wall is introduced and investigated. Upon varying a control parameter, the critical behaviour observed along the transition line changes from a directed-percolation to a multiplicative-noise type. Numerical simulations allow for a quantitative study of the multicritical point separating the two regions, Mean-field arguments and the mapping on a yet simpler model provide some further insight on the overall scenario.Comment: 4 pages, 3 figure

Yang-Lee zeros for a nonequilibrium phase transition

Equilibrium systems which exhibit a phase transition can be studied by investigating the complex zeros of the partition function. This method, pioneered by Yang and Lee, has been widely used in equilibrium statistical physics. We show that an analogous treatment is possible for a nonequilibrium phase transition into an absorbing state. By investigating the complex zeros of the survival probability of directed percolation processes we demonstrate that the zeros provide information about universal properties. Moreover we identify certain non-trivial points where the survival probability for bond percolation can be computed exactly.Comment: LaTeX, IOP-style, 13 pages, 10 eps figure

One-transit paths and steady-state of a non-equilibrium process in a discrete-time update

We have shown that the partition function of the Asymmetric Simple Exclusion Process with open boundaries in a sublattice-parallel updating scheme is equal to that of a two-dimensional one-transit walk model defined on a diagonally rotated square lattice. It has been also shown that the physical quantities defined in these systems are related through a similarity transformation.Comment: 8 pages, 2 figure