Multi-dimensional percolation


Percolation is concerned with the existence of an infinite path in a 
random subgraph of the lattice Z^D.
We can rephrase this as the existence of a Lipschitz embedding of the 
infinite line Z into the random subgraph.
What happens if we replace the line Z with another lattice Z^d?  I'll 
answer this for all values of the two dimensions
d and D, and the Lipschitz constant.  Based on joint works with Dirr, 
Dondl, Grimmett and Scheutzow.