## Branching random walk with catalysts.(English)Zbl 1064.60196

Summary: N. M. Shnerb, Y. Louzoun, E. Bettelheim and S. Solomon [Proc. Natl. Acad. Sci. USA 97, No. 19, 10322–10324 (2000; Zbl 0955.92001)] and N. M. Shnerb, E. Bettelheim, Y. Louzoun, O. Agam and S. Solomon [Phys. Rev. E 63, 021103 (2001)] studied the following system of interacting particles on $$\mathbb{Z}^d$$: There are two kinds of particles called $$A$$-particles and $$B$$-particles. The $$A$$-particles perform continuous time simple random walks independent of each other. The jumprate of each $$A$$-particle is $$D_A$$. The $$B$$-particles perform continuous time simple random walks with jumprate $$D_B$$, but in addition they die at rate $$\delta$$ and a $$B$$-particle at $$x$$ at time $$s$$ splits into two particles at $$x$$ during the next $$ds$$ time units with a probability $$\beta N_A(x,s)ds+ o(ds)$$, where $$N_A(x, s)$$ $$(N_B(x,s))$$ denotes the number of $$A$$-particles (respectively $$B$$-particles) at $$x$$ at time $$s$$. Conditionally on the $$A$$-system, the jumps, deaths and splittings of different $$B$$-particles are independent. Thus the $$B$$-particles perform a branching random walk, but with a birth rate of new particles which is proportional to the number of $$A$$-particles which coincide with the appropriate $$B$$-particles. One starts the process with all the $$N_A(x,0)$$, $$x\in \mathbb{Z}^d$$, as independent Poisson variables with mean $$\mu_A$$, and the $$N_B(x, 0)$$, $$x\in\mathbb{Z}^d$$, independent of the $$A$$-system, translation invariant and with mean $$\mu_B$$.
Shnerb et al. (2000) made the interesting discovery that in dimensions 1 and 2 the expectation $$\mathbb{E}\{N_B(x,t)\}$$ tends to infinity, no matter what the values of $$\delta,\beta,D_A$$, $$D_B,\mu_A, \mu_B\in(0,\infty)$$ are. We shall show here that nevertheless there is a phase transition in all dimensions, that is, the system becomes (locally) extinct for large $$\delta$$ but it survives for $$\beta$$ large and $$\delta$$ small.

### MSC:

 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60J80 Branching processes (Galton-Watson, birth-and-death, etc.)

### Keywords:

survival; extinction

Zbl 0955.92001
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