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, 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

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