Consider a random walk along nearest neighbors on an infinite, periodic face centered two-dimensional (d = 2) square lattice. A unit cell is shown in Fig. 4.19. Assume that the lattice spacing is a = 2 and that the walker starts at site = 0,2 = 0.

(a) By counting paths (draw them), find the probability, P4(0), that the walker returns to the origin after four steps. Find the probability, Q4(0), that the walker returns to the origin for the first time after four steps.

(b) Compute the generating function, U(z, 0). Use this result to compute P4(0) and Q4(0).

(c) Compute the escape probability.