Size doubling amoeba

Question

Note: Here time is assumed to be discrete and indexed by the naturals $\mathbb N$.

A curious species of amoeba undergoes the following evolution rule - at each time step, with probability $0 < p < 1$ it splits into two copies of itself of size $\varepsilon S$, where $S$ is the original size of the amoeba, and $0 < \varepsilon < 1$ is a fixed number. With probability $q = 1-p$, it doubles in size instead.

Start with one amoeba of size $1$. Denoting by $S^{\max}_t$ the size of the largest amoeba at time $t$, for what values of the parameters $\varepsilon, p$ do we have

$$\lim_{t \to \infty} S^{\max}_t = \infty$$

almost surely?

Answer 1

This question can be cast in the language of branching random walks: at each (discrete) time, a particle splits with probability $p$ and each of the children moves $\log \epsilon$; if it did not split, it moves $\log 2$. You are asking about the largest particle moving to infinity, i.e. the velocity being positive (essentially). This fits the model described e.g. in Zhan Shi's St Flour lecture notes, specifically Theorem 2.3 there. Let $\psi(t)= \log (2p \epsilon^t+q 2^t)$, then $v=\inf \psi(t)/t$, and you are asking whether $v>0$ or not. This is the sought after criterion.

The link to Shi's lecture notes is: https://link.springer.com/book/10.1007/978-3-319-25372-5