Branching Random Walk with Catalysts
Harry Kesten, Cornell University
Vladas Sidoravicius, IMPA
Abstract
Shnerb et al. (2000), (2001) studied the following system of interacting
particles on Zd: There are two kinds of particles, called
A-particles and B-particles. The A-particles perform continuous
time simple random walks, independently of each other.
The jumprate of each A-particle is DA. The B-particles
perform continuous time simple random walks with jumprate DB,
but in addition they die at rate δ and a B-particle at x
at
time s splits into two particles at x during the next
ds time
units with a probability β NA(x,s) ds
+o(ds), where NA(x,s)(NB(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 NA(x,0), x in
Zd, as independent
Poisson variables with mean μA, and the
NB(x,0), x in Zd,
independent of the A-system,
translation invariant and with mean μB.
Shnerb et al. (2000) made the interesting discovery that in
dimension 1 and 2 the expectation E{NB(x,t)} tends to
infinity, no
matter what the values of δ, β, DA,
DB , μA,μB
in (0,∞) are. We shall show here that nevertheless
there is a phase transition in all dimensions, that is, the
system becomes (locally) extinct for large δ but it survives for
β large and δ small.
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