is therefore natural to ask how close the law $\nu_T$ of the path chosen by the invasion cluster is to $\mu_T$. Showing that it has the right entropy is not too involved. This is Theorem \[th:small\] below. It is also easy to see that the two laws are typically not equal. As an example, consider the set of trees with first three generations given by

![image](figure.pdf)

When averaged over the remaining generations with the Galton-Watson measure—or equivalently, placing independent Galton-Watson trees at the terminal nodes—the limit-uniform measure splits equally at the root, while the invasion measure favors the left subtree regardless of the offspring distribution.

The best comparison one might hope for is that $\nu_T$ be absolutely continuous with respect to $\mu_T$, perhaps even with Radon-Nikodym derivative in $L^p$. Our main result is as follows.

\[th:main\] Suppose the offspring distribution $Z$ has at least $p$ moments, set $p_1 := \P[Z = 1]$, let $\mu:= \E[Z]$, and denote $q:=\frac{\log\mu}{\log\left(1/p_1\right)}$. If $$2p^2 q^2+(3p^2+5p)q+(-p^2+11p-4)<0,$$ then $\nu_\TT \ll \mu_\TT$ almost surely.

The condition in Theorem \[th:main\] is a trade-off between $p_1$ and $p$. In the case of $p = \infty$, the condition becomes $p_1 < 1/\mu^{\frac{3+\sqrt{17}}{2}}$. In the case of $p_1 = 0$, the 