3 FIG. 1: The survival probability vs. time for N = 1, 2, 3, and 4, together with power-law fits for the cases N = 1 and N = 4. attempted return to the origin and so the critical p has to be larger than 1/2 to compensate. The change in ex- ponent is not unexpected. For directed percolation, the critical exponents in a model of unrestricted occupancy are the mean-field exponents, which differ from the ex- ponents in the restricted occupancy version[18]. In the case of directed percolation, for any finite N the critical exponents are those of the N = 1 model. For large N , there is a crossover in the scaling between the mean-field scaling of the N = ∞ model and the directed percolation exponents [18]. Based on this analogy, one would expect a similar behavior in the present case, with the critical behavior for any finite N being identical to the N = 1 model. To test this hypothesis, we have measured the survival exponent δ at criticality for various N . The results are shown in Fig. 1. We see that, contrary to our naive expectation, there is a different exponent for each N . The measured best fit exponents are, for example, δ = 0 . 160 for N = 1, δ = 0 . 171 for N = 2, δ = 0 . 180 for N = 3 and δ = 0 . 192 for N = 4. There is no sign of any crossover behavior. One test for this is shown in Fig. 2, where we show P$_{S}$ ( t ) t $^{δ}$, in a linear( y )-log( x ) scale. We see that the graphs all exhibit the same characteristic oscillation in ln( x ) [19] that makes accurate determination of an exponent so difficult, but they show no secular trend. Varying N is not the only way to interpolate between the two models and thereby vary δ . We can instead intro- duce a parameter Δ which controls the lifetime of a par- ticle on the active site after reproduction (in the original model, Δ = 1). As Δ decreases, the interference between different particles is reduced, and the theory approaches the noninterference, Galton-Watson, limit. We can also Thus, in the case of normal diffusion, i.e. β = 1 / 2, for our modified model with unrestricted occupation of the origin, we have a phase transition at p = 1 / 2 with a critical exponent for the survival probability of δ = 1 / 4, observed also in numerical simulations [2] . This is in contrast with the original model, with a larger critical p , but more importantly, a survival exponent of δ ≈ 0 . 17. The change in the phase transition point is clear, since in the original N = 1 model, some children die upon their Thus for the survival probability of the process, P$_{S}$ ( t ) = 1 − P$_{E}$ ( t ), we obtain the power-law behavior P$_{S}$ ( t ) ∼ t − δ with δ = β/ 2. The special properties of β = 1 / 2 are not observed in the limit of s ≫ t , and as already been mentioned no aging behavior could be observed for a one-time quantity. The noncomutativity of the limits t → ∞ - s → ∞ is similar in some sense to the non commutativity of the same limits present for correlation function behavior in glassy systems [4] and leads to such non-equilibrium property as ergodicity breaking [16, 17] P$_{E}$ ( t ) ∼ 1 −$^{(}$ ψ$_{0}$ β ) β/ 2 t − β/ $^{2}$. (14) A ( s ) =$^{{}$ 1 − p 1 − 2 p ψ$_{0}$ β p < 1 / 2 1 p 1 − 2 p ψ$_{0}$ β p > 1 / 2 . (13) α = β, (12) and The transition in this limit is different then in the pre- viously discussed limit of t ≫ s , as A diverges similarly near p$_{c}$ and thus describes the crossover time scale on which one would observe critical behavior. For the criti- cal state p = p$_{c}$ we obtain Now we treat the limit s ≫ t ≫ 1, where we take first s → ∞ . In this limit we can neglect the second term on the right hand side of Eq. (3) and define P$_{E}$ ( t, ∞ ) = P$_{E}$ ( t ) simply as the extinction probability of the process. We are now dealing with a single-time quantity and so no aging behavior could be obtained. For the solution we use the same ansatz as in Eq. (4), writing A ( ∞ ) = A . The equation for P ∞ E remains the same as Eq. (5) and one again obtains the same critical value for p , p$_{c}$ = 1 / 2. In the off-critical state the results for α and A are times this is usually defined as aging, as in our case P$_{S}$ ( t, s ) ∝ ( s/t ) 1 − $^{β}$. The aging behavior for β ≤ 1 / 2 at criticality is a signature of the nonequlibrium phase tran- sition properties, which have been studied extensively in the context of DP and Contact Process [5, 14, 15], and the nonstationarity of the process usually obtained in glassy dynamics [4]. We must note that the obtained re- sult where the probability to observe at least one particle in time interval s is proportional to s 1 − β ( β < 1 / 2) is un- expected in light of the fact that for the off-critical state ( p < p$_{c}$ ), this probability is proportional to s . 10 0 10 4 10 8 10 12 10 16 t 0.001 0.01 0.1 1 P S N N N N t t