EXERCISE 5.4 Consider an asymmetric random walk on a lattice with five lattice sites. Assume that the fifth site, P, absorbs the walker. The transition rates w1,2 = W1,3 = W1,4 = W1, p = , W2,1 = W2,3 = W2, p = , W3,1 = W3,2 = W3,4 = , W4,1 = W4,3 = w4, p = , WP,1 = WP,2 = WP,4 and wij 0 for all other transitions.

(a) Write the transition matrix, M, and compute its eigenvalues and left and right eigenvectors.

(b) If the walker starts at site n = 3 at time t = 0, compute the mean first passage time.