Consider a random walk along nearest neighbors on an infinite, periodic body-centered three-dimensional (d = 3) cubic lattice. A unit cell is shown in Fig. 4.20. Assume that the lattice spacing is a = 2 and that the walker starts at site,

(1=0,2 = 0,3 = 0). Compute the generating function. U(1,0), and the escape probability. [Hint: Make a change of variables, x = tan(o/2). Then change to spherical coordinates, x = rsin(0)cos(o), x2 = rsin(0) sin(), and x = rcos(0). Replace the integration over by an integration over I, where t = rsin(0) cos(0) sin(). The three integrations over t, 0, and then separate.]

Transcribed Image Text:

a a Fig. 4.19. Unit cell for a face-centered square lattice.