We present a Monte Carlo study in dimension d=1 of the two-species reaction-diffusion process A+B-->2B and B-->A. Below a critical value rho(c) of the conserved total density rho the system falls into an absorbing state without B particles. Above rho(c) the steady state B particle density rho(st)(B) is the order parameter. This system is related to directed percolation but in a different universality class identified by Kree et al. [Phys. Rev. A 39, 2214 (1989)]. We present an algorithm that enables us to simulate simultaneously the full range of densities rho between zero and some maximum density. From finite-size scaling we obtain the steady state exponents beta=0.435(10), nu=2.21(5), and eta=-0.606(4) for the order parameter, the correlation length, and the critical correlation function, respectively. Independent simulation indicates that the critical initial increase exponent takes the value straight theta(')=0.30(2), in agreement with the theoretical relation straight theta(')=-eta/2 due to Van Wijland et al. [Physica A 251, 179 (1998)].

We investigate percolation phenomena in multifractal objects that are built in a simple way. In these objects the multifractality comes directly from the geometric tiling. We identify some differences between percolation in the proposed multifractals and in a regular lattice. There are basically two sources of these differences. The first is related to the coordination number, which changes along the multifractal. The second comes from the way the weight of each cell in the multifractal affects the percolation cluster. We use many samples of finite size lattices and draw the histogram of percolating lattices against site occupation probability. Depending on a parameter characterizing the multifractal and the lattice size, the histogram can have two peaks. We observe that the percolation threshold for the multifractal is lower than that for the square lattice. We compute the fractal dimension of the percolating cluster and the critical exponent beta. Despite the topological differences, we find that the percolation in a multifractal support is in the same universality class as standard percolation.

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Recent Monte Carlo results ͓Phys. Rev. E 61, 6330 ͑2000͔͒ on the one-dimensional reaction-diffusion process AϩB→2B and B→A lead us to estimate ϭ2.21Ϯ0.05 for the correlation length exponent. The preceding Comment ͓Phys. Rev. E 64, 058101 ͑2001͔͒ advocates the exact value ϭ2. We show that Janssen's arguments leave enough doubts to justify an independent Monte Carlo determination of .In our work ͓1͔ referred to by the preceding Comment ͓2͔, we treated the exponent as an unknown, with the result ϭ2.21Ϯ0.05; any other value of deteriorates the fit to the order parameter curve, based on our Fig. 3 and the discussion of Sec. III A 2. We should have mentioned the relation ϭ2/d due to Kree, Schaub, and Schmittmann ͑KSS͒ ͓3͔, and commented on it not being satisfied in our dϭ1 simulation.Now the Comment asserts ͑although with a proviso: ''as long as one assumes . . . ''͒ that ϭ2 has to be inserted in the analysis of dϭ1 Monte Carlo data as if it were an established fact. ͓Anyone convinced of that only has to solve our equation ␤/ϭ0.197Ϯ0.002 ͓1͔ with ϭ2 ͑instead of ϭ2.21Ϯ0.05) to find ␤ϭ0.394Ϯ0.004 ͑instead of ␤ ϭ0.435Ϯ0.010); thereby sacrificing the fit to the orderparameter curve. ͔ We show below that the Comment is misguided in attempting to impose ϭ2 as an a priori truth, not in need of independent verification. The author's criticism of our exponent values on this basis is therefore inadmissible.Our value of was stated ͓1͔ not to take into account any possible systematic corrections. One is free to hope that if and when such corrections can be handled, they will reduce the Monte Carlo value of to 2. Alternatively, this may be a case where Janssen's arguments ͓2͔ do not apply and where 2.A good reason for their potential failure, and hence for circumspection, is of a well-known kind. The relation ϭ2/d depends on the existence of a fixed point having all the symmetries listed in the Comment. The existence of such a fixed point was demonstrated in an ⑀ expansion ͑the ''KSS fixed point'' ͓3͔͒, but is increasingly subject to doubt as d is lowered: The quartic terms in the action ͑see ͓4͔͒, which are irrelevant in the ⑀ expansion, may become relevant for d as far down as dϭ1. A warning signal ͑not a proof͒ is that at the Gaussian fixed point these terms are ''naively relevant'' for dϽ2, and that at the KSS fixed point, due to being negative, their scaling dimensions increase with ⑀.If the quartic terms are relevant in dϭ1, then, because they break the continuous symmetry on which ϭ2 is based, these terms destroy this relation. The discrete symmetries ͑IV and V in the Comment's notation͒, however, continue to be respected by the quartic terms ͓4͔, and it is hard to see how, even if they are relevant, these terms could affect the exponent relations that we used later on in our work ͓1͔. In particular, ϭ is guaranteed by the time-reversal symmetry of the action ͓4͔, quartic terms included. KSS define their model originally as a system of two coupled Langevin equations ͓ ͓3͔, Eqs. ͑2.1b͒ and ͑2.4͒-͑2.6͔͒, which leave the quar...