condition is $p>\frac{11+\sqrt{105}}{2}$.

An outline of the argument behind Theorem \[th:main\] is as follows. Let $X_n$ be the KL-distance between the way that $\mu_T$ and $\nu_T$ split at the $n$th step $\gamma_n$ of a path chosen from $\nu_T$. A sufficient condition for absolute continuity is that $\sum_{n=1}^\infty \E X_n < \infty$. A precise statement is given in Lemma \[lem:X\] below. A more detailed outline of why this condition should hold is given at the beginning of Section \[sec:rest\].

The reason we have a hope of estimating $X_n$ is that there is a [*backbone*]{} decomposition for invasion percolation. Define the backbone to be the almost surely unique nonbactracking path $\gamma = (\rtt, \gamma_1, \gamma_2, \ldots)$ from the root to infinity. For any vertex $v$ define the [*pivot value*]{} at $v$, denoted $\pivot (v)$ \[pivot-def\], to be the least $p$ such that there is a path from $v$ to infinity in the subtree at $v$ with all $U$ variables (not including the one at $v$) at most $p$. On a regular tree, invasion percolation was studied in [@angel-invasion2008; @angel-invasion2013]. For the purposes of studying $\nu_T$, the regular tree is a degenerate case, because $\mu_T$ and $\nu_T$ are equal to each other and 