Abstract Exact results are presented for conditioned dynamics in a system of interacting random walks in one dimension that annihilate immediately when two particles meet on the same site and where pairs of particles are deposited randomly on neighbouring sites. For an atypical hopping activity one finds dynamical nonequilibrium phase transitions analogous to the zero-temperature equilibrium phase transitions that appear in the spin-1/2 quantum XY spin chain in a transverse magnetic field. Along the critical line the approach of the particle density to its stationary value is algebraic with an unexpected mean field exponent. The time-dependent local stationary density correlations are universal with dynamical exponent z = 1. Inside the disordered phase spatially oscillating correlations appear below the typical activity.
Using Monte Carlo simulations we have studied the transition from an ``active'' steady state to an absorbing ``inactive'' state for two versions of the branching annihilating random walks with parity conservation on a square lattice. In the first model the randomly walking particles annihilate when they meet and the branching process creates two additional particles; in the second case we distinguish particles and antiparticles created and annihilated in pairs. Quite distinct critical behavior is found in the two cases, raising the question of what determines universality in these kinds of systems.
We study the effects of hard core (HC) interactions between different species of particles on two-species branching annihilating random walks with one offspring(BAW
2(1)). The single-species model belongs to the directed percolation (DP) universality class. In the BAW
2(1) model, a particle creates one particle of the same species in its neighborhood with the probability
σ(1−p) and of the different species with
(1−σ)(1−p), where
p is the hopping probability. Without HC interactions, this model always exhibits the DP-type absorbing transition for all
σ. Even with HC interactions, the nature of the phase transitions does not change except near
σ=0, where the HC interaction destabilizes and completely wipes away the absorbing phase. The model is always active except at the annihilation fixed point of zero branching rate (
p=1). Critical behavior near the annihilation fixed point is characterized by exponents
β=ν⊥=1/2 and
ν∣∣=1.
We study a d-dimensional system of diffusing particles that on contact either annihilate with probability $1/(q\ensuremath{-}1)$ or coagulate with probability $(q\ensuremath{-}2)/(q\ensuremath{-}1).$ In one dimension, the system models the zero-temperature Glauber dynamics of domain walls in the q-state Potts model. We calculate
P(m,t), the probability that a randomly chosen lattice site contains a particle whose ancestors have undergone exactly $(m\ensuremath{-}1)$ coagulations. Using perturbative renormalization group analysis for
d<2, we show that, if the number of coagulations m is much less than the typical number
M(t), then $\overline{P}(m,t)\ensuremath{\sim}{m}^{\ensuremath{\zeta}/d}{t}^{\ensuremath{-}\ensuremath{\theta}},$ with $\ensuremath{\theta}=dQ+Q(Q\ensuremath{-}1/2)\ensuremath{\epsilon}+O({\ensuremath{\epsilon}}^{2}),$ $\ensuremath{\zeta}=(2Q\ensuremath{-}1)\ensuremath{\epsilon}+(2Q\ensuremath{-}1)(Q\ensuremath{-}1)(1/2+AQ){\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3}),$ where $Q=(q\ensuremath{-}1)/q,$ $\ensuremath{\epsilon}=2\ensuremath{-}d$ and $A=\ensuremath{-}0.006\dots{}.$
M(t) is shown to scale as $M(t)\ensuremath{\sim}{t}^{d/2\ensuremath{-}\ensuremath{\delta}},$ where $\ensuremath{\delta}=d(1\ensuremath{-}Q)+(Q\ensuremath{-}1)(Q\ensuremath{-}1/2)\ensuremath{\epsilon}+O({\ensuremath{\epsilon}}^{2}).$ In two dimensions, we show that $\overline{P}(m,t)\ensuremath{\sim}\mathrm{ln}{(t)}^{Q(3\ensuremath{-}2Q)}\mathrm{ln}{(m)}^{(2Q\ensuremath{-}{1)}^{2}}{t}^{\ensuremath{-}2Q}$ for $m\ensuremath{\ll}{t}^{2Q\ensuremath{-}1}.$ We also derive an exact nonperturbative relation between the exponents: namely $\ensuremath{\delta}(Q)=\ensuremath{\theta}(1\ensuremath{-}Q).$ The one-dimensional results corresponding to $\ensuremath{\epsilon}=1$ are compared with results from Monte Carlo simulations.
We present some exact results on the behavior of branching and annihilating random walks, both in the directed percolation and parity conserving universality classes. Contrary to usual perturbation theory, we perform an expansion in the branching rate around the nontrivial pure annihilation (PA) model, whose correlation and response function we compute exactly. With this, the nonuniversal threshold value for having a phase transition in the simplest system belonging to the directed percolation universality class is found to coincide with previous nonperturbative renormalization group (RG) approximate results. We also show that the parity conserving universality class has an unexpected RG fixed point structure, with a PA fixed point which is unstable in all dimensions of physical interest.
The effect of blocking between different species occurring in one dimension is investigated here numerically in the case of particles following branching and annihilating random walk. It is shown that two-dimensional simulations confirm the field theoretical results with logarithmic corrections. In one dimension, however, if particles exhibit hard core interaction I confirm the very recent predictions of Kwon et al. [Phys. Rev. Lett. 85, 1682 (2000)] that there are two different universality classes depending on the spatial symmetry of the offspring production characterized by beta(S)=0.5 and beta(A)=2. Elaborate analysis of simulation data shows that the order parameter exponent beta does not depend on initial conditions or on diffusion rates of species but strong correction to scaling is observed. By systematic numerical simulations the critical point properties have been explored and initial condition dependence of the dynamical exponents Z and alpha is shown. In the case of a random initial state the particle-density decay at the critical point follows the t(-1/4) law with logarithmic corrections with two offsprings.