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**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.

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.) |