2 the boundary produces exactly one offspring. Below this point, the number of particles decreases exponentially with the number of past boundary particles, and above it it grows exponentially. The behavior of A ( s ) and α for the off-critical state, p = p$_{c}$ , is given by for the first particle [10], we obtain [11] the equation for G$_{(}$$_{t,t}$$_{+}$$_{s}$$_{)}$ ( s ) G$_{(}$$_{t,t}$$_{+}$$_{s}$$_{)}$ ( s ) = ∫ t 0 G$_{O}$ ( G$_{(}$$_{t}$$_{−}$$_{u,t}$$_{−}$$_{u}$$_{+}$$_{s}$$_{)}$ ( s )) ψ ( u ) du + s$^{∫}$ t + s t G$_{O}$ ( G$_{(0}$$_{,t}$$_{−}$$_{u}$$_{+}$$_{s}$$_{)}$ ( s )) ψ ( u ) du +$^{∫}$ ∞ t + s ψ ( u ) du. (2) A ( s ) = { 1 − p 1 − 2 p ψ$_{0}$ β s p < 1 / 2 − 1 p > 1 / 2 , (6) α = { 1 + β p < 1 / 2 β p > 1 / 2 . (7) In the limit of s → 0 the generating function goes to P ( Z ( t, t + s ) = 0) := P$_{E}$ ( t, s ), i.e. the probability that not a single particle appears on the boundary at the men- tioned time interval, taking this limit in Eq. (2) we obtain The presence of the phase transition as we approach p = 1 / 2 from below is clearly manifest in the divergence of the coefficient A . The approach from above is not obvious from the large t behavior. What happens is that for p ≳ 1 / 2, P$_{E}$ first rises toward unity, as happens below the transition. However, at very large t , the behavior crosses over toward the power-law decay toward P ∞ E . The details of this crossover will be presented in our longer presentation [11]. P$_{E}$ ( t, s ) = ∫ t 0 1 − p 1 − pP$_{E}$ ( t − u, s ) ψ ( u ) du +$^{∫}$ ∞ t + s ψ ( u ) du, (3) In the critical state, p = p$_{c}$ Eq. (5) remains valid and so P ∞ E = 1 while the behavior of A ( s ) and α shows a transition as a function of β . For α , we obtain α = { 1 − β β ≤ 1 / 2 β β ≥ 1 / 2 , (8) A ( s ) = − π csc( βπ )Γ(1 − b ) Γ(1 − 2 β )Γ(1 + β ) ( β > 1 / 2) (9) P$_{E}$ ( t, s ) = P ∞ E − A ( s ) t − α t → ∞ (4) A ( s ) ∝ s 1 − β ( β < 1 / 2) . (10) For β < 1 / 2, A ( s ) has to be calculated numerically in general. However, in the limit of large s , we have We can exhibit an analytic expression for A ( s ) at p$_{c}$ only for β > 1 / 2: The existence of a special β for the behavior at criticality is very non-trivial and it is especially interesting that the critical β$_{c}$ is equal to 1 / 2, i.e the normal diffusion case. We can treat this critical β$_{c}$ as a critical fractal dimension since 2 β is just the fractal dimension of the diffusion. For the special case of β = 1 / 2 our ansatz, i.e. Eq. (4), does not work and one needs to treat this case specially [11]; the result is 1 − P$_{E}$ ( t, s ) ∝ 1 log( s $^{/}$t ) $^{(}$s$^{/}$t) 1 / 2 ( β = 1 / 2) . (11) The logarithmic corrections in the behavior support our claims as to the critical nature of β = 1 / 2. For a two time quantity like the survival probability P$_{S}$ ( t, s ) = 1 − P$_{E}$ ( t, s ), one usually expects for a stationary pro- cess a dependence only on the time difference s ; when the process is non-stationary this is generally not true. When the time dependence is that of a ratio of the two which clearly points out the existence of a critical p = p$_{c}$ = 1 / 2. The location of the critical point at p = 1 / 2 is due to the fact that at this value, each particle on P ∞ E = { 1 − p p p ≥ 1 / 2 1 p ≤ 1 / 2 (5) In order to obtain a solution of Eq. (3) we substitute the asymptotic form in Eq. (4) into both sides of Eq. (3), perform an expansion in inverse powers of t and com- pare the coefficients in front of the appropriate leading terms. We leave the exact technical details for a longer publication [11] and now provide only the final results. So doing, in the limit t → ∞ , s fixed, we obtain We will now explore the behavior of P$_{E}$ ( t, s ) in the asymptotic limit of large t for different values of the pa- rameters p and β . Inspired by numerical solutions of Eq. (3), we adopt the ansatz where we have now used the explicit from of G$_{O}$ ( s ). Eq (3), a non-linear Volterra equation of the second kind, is our main equation for the unrestricted case of the model and it describes the time evolution of a two-time quantity. We haven’t assumed anything as to the form of ψ ( t ) and so Eq. (3) is quite general. The long-time behavior of P$_{E}$ is, as we shall see, governed entirely by the long time behavior of ψ ( t ). For normal diffusion in the bulk, this long-time behavior is ψ$_{t}$ ∼ ψ$_{∞}$t − (1+ β $^{)}$, with β = 1 / 2, ψ$_{∞}$ = 1 / √ 4 πD , where D is the diffusion constant (defined as usual). We also consider the more general case of 0 < β < 1, and in such case the con- stant ψ$_{∞}$ is used for the normalization of ψ$_{t}$ . The case of 0 < β < 1 can be achieved if the diffusion in the bulk is anomalous [12]; e.g., subdiffusion for 0 < β < 1 / 2 de- scribed by models such as the continuous time random walk (CTRW) [12, 13]. Herein we study P$_{E}$ exclusively; other quantities such as the mean number of particles can also be obtained from Eq. (2) [11].