Consider a Galton-Watson branching process, with offspring distribution $\mathbf{p}=(p_0, p_1, \dots, p_n, \dots)$. Let $O$ be the root of the branching process.

Write $\eta=P(\text{process survives for ever})$ and $\mathcal{H}=\{\mathbf{p}: \eta>0\}$.

Also write $\beta=P(O \text{ is the root of an infinite binary tree contained in the branching process})$ and $\mathcal{B}=\{\mathbf{p}: \beta>0\}$.

Then it's fairly well known that in some sense the phase transition from $\mathcal H^C$ to $\mathcal H$ is typically "continuous" while the phase transition from $\mathcal B^C$ to $\mathcal B$ is typically "discontinuous".

e.g. suppose the offspring distribution is Poisson with mean $\lambda$ and consider $\eta$ as a function of $\lambda$. Then $\eta(\lambda)=0$ for $\lambda\leq 1$ and $\eta(\lambda)>0$ for $\lambda>1$, and $\eta(.)$ is continuous everywhere, including at 1.

On the other hand, consider $\beta$ as a function of $\lambda$. Then there is a critical point $\lambda_c\approx 3.3509$ with the following property: $\beta(\lambda)=0$ for $\lambda<\lambda_c$, and $\beta(\lambda)>0$ for $\lambda\geq \lambda_c$. In particular, $\beta(.)$ is discontinuous at $\lambda_c$. (In fact, at $\lambda_c$, $\beta$ jumps from 0 to approximately 0.535).

My question: how has this been written as a general statement? (rather than just for particular parametrised families as above). My guess is that one would want to write it something like the following:

Let $M_0$ be the set of offspring distributions with the topology induced by the metric $d_0(\mathbf{p}, \mathbf{q})=\sum|p_n-q_n|$.

Similarly $M_1$ and $M_2$ induced by $d_1(\mathbf{p}, \mathbf{q})=\sum n|p_n-q_n|$ and $d_2(\mathbf{p}, \mathbf{q})=\sum n^2|p_n-q_n|$.

Write $\mathbf{p}^*$ for the degenerate distribution with $p^*_n=0$ for $n=0$ and $n\geq 2$ and $p^*_1=1$.

Then:

(1) $\mathcal{H}\setminus\{\mathbf{p}^*\}$ is open as a subset of $M_0$.

(2) $\mathcal{B}$ is closed as a subset of $M_2$. (Probably also $M_1$?)

Of course (1) is basically trivial, since $\mathcal{H}$ is just the set of distributions with mean greater than 1, along with the single point $\mathbf{p}^*$.

(Note that it's NOT the case that $\mathcal{B}$ is closed as a subset of $M_0$. For example, consider a sequence of distributions indexed by $k$ with $p_0=1-4/k$ and $p_k=4/k$, all other $p_n=0$. Then $\beta(k)>0$ for all $k$, but $\beta(k)\to 0$ as $k\to\infty$, and the sequence of distributions converges in $M_0$ to a limit in which just $p_0=1$, for which of course $\beta=0$.)

I don't think it's hard to write down a proof of (2) above, but I don't want to reinvent the wheel. (I don't need to use the result directly, but I would definitely like to refer to it to illustrate a point). So: anyone know where such things have been nicely developed before?