The random walk problem drunken sailor walk basic

The random walk problem (drunken sailor walk) basic probability theory --> http: //www-math. bgsu. edu/~albert/m 115/probability/outline. html (We need the concept of probability, probability distributions, calculating averages) ? ? ? The problem: There is a drunk man coming out of a pub. He cannot control his steps, and randomly (with 1/2 probability makes a step forward and backward, Let us assume that the length of his steps is fixed (1 m). A. Determine the probability, that after N steps he is at a distance L from the starting point. B. Determine <L 2>=f(N)=? C. How about 2 D and 3 D ? Representation of one possible track total number of tracks: Nt=2 N the probability to follow one given track is P 1=1/2 N

The number of tracks with N steps that are finishing at coordinate x --> W(N, x) = ? - possible values of x--> {N, N-2, N-4, ……-(N-2), -N} - let N+ be the number of steps in + direction; N- the number of steps in - direction N++N-=N; N+-N-=x; --> we get: The P(N, x) probability that after N steps the random walker is at coordinate x is: - Due to the presence of factorials it is hard to work with P(N, x) as given above. - A more analytical form can be obtained by using the Stirling formula: We get:

If N>>1 the important part of P(N, x) is for x<<N. We use thus the x/N<<1 simplification and write: After neglecting the second order terms in x/N we get: Which is normalized to 2, on [- , ]-since it is valid for only each second integer x [otherwise P(N, x)=0] We can calculate now <x>, and <x 2>: Generalization in 2 d and 3 d (square and cubic lattice sites) a random walk of N steps in 2 d --> a random walk of N/2 steps along the x axis + a random walk of N/2 steps along the y axis for 3 d in the same manner:

Suggested further “research” 1. Prove (by computer simulations or analytically if you can…) that 3 d is the lowest dimensional space where somebody can get completely lost. . . I. e. the probability that the r. w. track crosses the starting point is going to zero, when the number of steps is going to infinity…. What will happen in 1 d and 2 d? (use square and cubic lattice sites in 2 d and 3 d, respectively) 2. Study by computer simulations the self-avoiding random walk. The self-avoiding random walk is a random walk, where the walker cannot move on a site previously visited. What will one expect in this case for the coefficient in 1 d, 2 d and 3 d? 3. Study by computer simulations the coefficient of a 2 d Levy-flight. The Levy flight is a random walk consisting of random jumps. The probability of jumping to a site at distance s, from the original site is decreasing as a power low with s and does not depend on the chosen direction. (use a square lattice)