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Description:
We study the scaling limit for a catalytic branching particle system whose particles performs random walks on $\ZZ$ and can branch at 0 only. Varying the initial (finite) number of particles we get for this system different limiting distributions. To be more specific, suppose that initially there are $n^{\be}$ particles and consider the scaled process $Z^n_t(\bullet)=Z_{nt}(\sqrt{n}\, \bullet)$ where $Z_t$ is the measure-valued process representing the original particle system. We prove that $Z^n_t$ converges to 0 when $\be<\frac{1}{4}$ and to a nondegenerate discrete distribution when $\be=\frac{1}{4}$. In addition, if $\frac{1}{4}<\be<\frac{1}{2}$ then $n^{-(2\be-\frac{1}{2})}Z^n_t$ converges to a random limit while if $\be>\frac{1}{2}$ then $n^{-\be}Z^n_t$ converges to a deterministic limit. Note that according to Kaj and Sagitov \cite{KS} $n^{-\frac{1}{2}}Z^n_t$ converges to a random limit if $\be=\frac{1}{2}.$