[@randall-sinclair], to determine the growth rate of the number of self-avoiding paths.

A number of methods have been studied. One is to do a random walk on the tree, with a “homesickness” parameter determining how much steps back toward the root are favored [@LPP-biased]. The parameter needs to be tuned near criticality: too much homesickness and the walk gets stuck near the root; too little homesickness and the walk goes to infinity without taking the time to ensure that the path is well randomized. Randall and Sinclair [@randall-sinclair] solve this by estimating the critical parameter as the walk progresses, re-tuning the homesickness to lie above this by an amount decreasing at an appropriate rate.

Another approach is to use percolation. One conditions the percolation cluster to survive to level $N$; as the percolation parameter decreases to criticality and $N$ is taken to infinity, the law of this cluster approaches the law of the [*incipient infinite cluster*]{} (IIC). For many graphs—e.g. regular or Galton-Watson trees—the IIC almost surely contains a unique infinite path, thereby giving a mechanism for sampling such a path. In practice, the same considerations arise as with homesick random walks: tuning the percolation parameter too low yields too 