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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.

MSC:

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

Keywords:

survival; extinction

Citations:

Zbl 0955.92001
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