  Let $G$ be a graph in which each vertex initially has weight 1. In each step,
the weight from a vertex $u$ can be moved to a neighbouring vertex $v$,
provided that the weight on $v$ is at least as large as the weight on $u$. The
total acquisition number of $G$, denoted by $a_t(G)$, is the minimum possible
size of the set of vertices with positive weight at the end of the process.
LeSaulnier, Prince, Wenger, West, and Worah asked for the minimum value of
$p=p(n)$ such that $a_t(\mathcal{G}(n,p)) = 1$ with high probability, where
$\mathcal{G}(n,p)$ is a binomial random graph. We show that $p = \frac{\log_2
n}{n} \approx 1.4427 \ \frac{\log n}{n}$ is a sharp threshold for this
property. We also show that almost all trees $T$ satisfy $a_t(T) = \Theta(n)$,
confirming a conjecture of West.
