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Non-Universal Extinction Transition for Boundary Active Site

We present a generalized model of a diffusion-reaction system where the reaction occurs only on the boundary. This model reduces to that of Barato and Hinrichsen when the occupancy of the boundary site is restricted to zero or one. In the limit when there is no restriction on the occupancy of the boundary site, the model reduces to an age dependent Galton-Watson branching process and admits an analytic solution. The model displays a boundary-induced phase transition into an absorbing state with rational critical exponents and exhibits aging at criticality below a certain fractal dimension of the diffusion process. Surprisingly the behavior in the critical regime for intermediate occupancy restriction $N$ varies with $N$. In fact, by varying the lifetime of the active boundary particle or the diffusion coefficient in the bulk, the critical exponents can be continuously modified.