  Invasion percolation is a stochastic growth model that follows a greedy
algorithm. After assigning i.i.d. uniform random variables (weights) to all
edges of $\mathbb{Z}^d$, the growth starts at the origin. At each step, we
adjoin to the current cluster the edge of minimal weight from its boundary. In
'85, Chayes-Chayes-Newman studied the `acceptance profile' of the invasion: for
a given $p \in [0,1]$, it is the ratio of the expected number of invaded edges
until time $n$ with weight in $[p,p+\text{d}p]$ to the expected number of
observed edges (those in the cluster or its boundary) with weight in the same
interval. They showed that in all dimensions, the acceptance profile $a_n(p)$
converges to one for $p<p_c$ and to zero for $p>p_c$. In this paper, we
consider $a_n(p)$ at the critical point $p=p_c$ in two dimensions and show that
it is bounded away from zero and one as $n \to \infty$.
