## The low-density limit of coalescing branching random walk

This is one of the "something like this must be true" situations.
The issue is to find a reasonably "clean" proof technique that can be adapted to both models below,
and hopefully to other variants too.
For concreteness take the lattice $Z^3$ -- what's important is random walk is transient.

** Model 1.**
Consider a model BCRW of branching coalescing random walk.
To be definite (though the point is that the details shouldn't matter) take particles behaving as:

(i) continuous-time simple RW

(ii) splitting at rate $\beta$, a daughter particle placed at an adjacent site

(iii) two particles coalesce into one if they meet.

Such a process has a (translation-invariant) stationary distribution for which the mean number of particles per site (*intensity*)
equals some $p(\beta)$.
See this Athreya-Swart paper for discussion.
There seems no simple exact formula for the complete stationary distribution, but one can calculate $p(\beta) \sim c \beta$ as $\beta \to 0$.

** Model 2.**
This is the constrained Ising model from
page 16-17 of The constrained
Ising model as an algorithm for storage in dynamic graphs
but now with $Z^3$ instead of a finite graph.
The model has a parameter $p$ and in the stationary distribution,
site are occupied as an i.i.d. Bernoulli(p) process.
As commented in the linked material,
this process evolves in a way that emulates some variant of CBRW.

**Problem.**
The issue is to show that, for Models 1 and 2 and related models where there is a stationary distribution,
as the intensity (which depends on model parameters) tends to 0, the suitable time-rescaled process
in which we track the ``lineages" (splits and merges) but not the spatial positions, of the particles, converges in distribution to
the following simple limit process.

**A limit process.**
The limit process is constructed w.r.t. a reference particle at time 0.
This has a lineage from the infinite past to the infinite future; ``splits" and ``merges" with some other lineage both occur
at times of Poisson(1) point processes. A split [merge] creates a new lineage running froward [backward], and recursively such a lineage
has its own splits and merges, each creating a new lineage.

In the conjectured convergence result we need to overlook "jitter", the times where a daughter particle soon coalesces with its parent, which we do by
asking only for convergence of finite dimensional distributions (in time).