little likelihood of survival and too great a time cost to rejection sampling; too great a percolation parameter results in too many surviving paths and a selection problem which leads to poor randomization.

Invasion percolation was introduced as a model for how viscous fluid creeps through an environment in [@invasion]. Each site is given an independent $U[0,1]$ random variable, representing how great the percolation probability would have to be before the site would be open. The cluster then grows by adding, at each time step, the site with the least $U$ value among sites neighboring the cluster but not in the cluster. It is not hard to see that the $\limsup$ of $U$-values of bonds chosen is equal to the critical percolation parameter. In other words, instead of running percolation at $p_c$ and conditioning to survive, one allows slightly supercritical bonds but less and less as the cluster grows. As is the case for the IIC, the invasion cluster almost surely contains only one infinite path in the case of regular or Galton-Watson trees, and thus gives a different mechanism for sampling paths. Unlike the IIC and homesick random walk, invasion percolation requires no tuning to criticality and is an 