2.1 Derive upper and lower bounds for the critical value p _c for site percolation on the square lattice following the outline of the proof of Theorem 2.2.5.

2.2 Find graphs with bond percolation critical value equal to zero and one respectively.

2.3 Find a graph with 0 < = < 1.

2.4 Prove the following statement (first convince yourself that there is actually something to prove at all!): in percolation on the two-dimensional integer lattice the origin is in an infinite component if and only if there is an open path from the origin to infinity.

2.5 Consider bond percolation on the one-dimensional line, where each edge is deleted with probability p = 1/2. Consider a segment of n edges. What is the probability that the vertices at the two sides of the segment are connected ? What happens if n increases? Consider now bond percolation on the square lattice again with p = 1/2. Consider a square with n vertices at each side. What is the probability that there exists a path connecting the left side with the right side?

2.6 In the proof of Theorem 2.2.7 we have compared the dynamic marking procedure with site percolation, and argued that if P( C = ?) > 0 in the site percolation model, then this is also true for the tree obtained using the marking procedure. Note that if we could argue the same for bond percolation,...