Problem 5.5. Consider a discrete random walk on a one-dimensional periodic lattice with 2N+1 lattice sites (label the sites from -N to M). Assume that the walker is equally likely to move one lattice site to the left or right at each step. Treat this problem as a Markov chain.

(a) Compute the transition matrix, Q, and the conditional probability matrix, P (sols).

(b) Compute the probability P(n, s) at time s, given the walker starts at site n = 0.

(c) If the lattice has five lattice sites (N = 2), compute the probability to find the walker on each site after s = 2 steps and after soo steps. Assume that the walker starts at site n = 0.