Recurrent and Nonrecurrent Random Walks Revisited (a) Consider a possibly asymmetric random walk in one dimension. A random walker starts at x = 0, and at every time step, the walker takes a step to...

Recurrent and Nonrecurrent Random Walks Revisited (a) Consider a possibly asymmetric random walk in one dimension. A random walker starts at x = 0, and at every time step, the walker takes a step to the right with probability p and a step to the left with probability q = 1- p , as illustrated in Fig. 2.5. Determine whether the random walk is recurrent. Recall that a random walk is said to be recurrent if it eventually returns to the origin with probability 1. Hint: Prove first that a random walk is recurrent if and only if the sum is finite. Here pn 0 is the probability of the walker being at x = 0 at time n . (b) Determine whether a symmetric random walk in d dimensions is recurrent. Hint: Since the exact expression for pn 0 is quite complicated for d > 1, you may wish to use an approximation. Problem credit: Pétur Rafn Bryde.