Random Walks Charles N. Moore Department of Mathematics, Kansas State University Manhattan, KS 66506 U.S.A. Abstract. We discuss the classical theorem of P´olya on random walks on the integer lattice in Euclidean space. This is the starting point for much work that has been done on random walks in other settings. We mention a tiny fraction of this work, and discuss in detail “random walks” which can be created using trigonometric functions. 1 P´ olya’s Theorem Consider the integers {..., −2, −1, 0, 1, 2,... }. Suppose you are standing at 0. Flip a coin. If the coin comes up heads, move to the right by one step. If it comes up tails, move to the left by one step. Flip the coin again. If it comes up heads, move a step to the right, if it comes up tails, move a step to the left. Repeat and continue this process. Must you come back to zero? Figure 1: Georg P´olya Obviously, there is a very good chance of returning to zero: the first two flips could have been head then tail, or tail then head, and these both result in a return to zero. So of the four possible outcomes in the first two flips, half of these take you back to zero. So the probability of returning to zero is at least 1 2 . If you further consider that even if you fail to return to zero in two steps, that after some subsequent flips you might make it back to zero, then clearly the probability of returning to zero is greater than 1 2 . A natural question arises: What is the probability of returning to zero? The answer was given by Georg P´olya [9] in 1921: Theorem 1. With probability one, the random walker will return to zero in a finite number of steps. Let us show how to prove P´olya’s theorem. For n =0, 1, 2, 3,... let u n = the probability that the random walker is at 0 after n steps. We make two simple observations: u 0 = 1 (because the walker starts at 0) and u n = 0 if n =1, 3, 5, 7,..., that is, the walker cannot be at 0 after an odd number of steps. For n =1, 2, 3,..., set f n = the probability that the walker returns to 0 for the first time at step n. Set f 0 = 0 (being at 0 at step 0 isn’t really a return) and f = ∞  n=0 f n . The probability that the walker returns to 0 is f , the probability of not returning is 1 − f. We now decompose the event that the walker is at 0 at time n, n ≥ 1, into first returns, that is, if the walker is at 0 at time n, then the walker may be back there for the first time or may have 1