The random walk problem (drunken sailor walk) basic probability theory --> http://www-math.bgsu.edu/~albert/m115/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 <L2>=f(N)=? C. How about 2D and 3D ? Representation of one possible track total number of tracks: Nt=2N the probability to follow one given track is P1=1/2N

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 <x2>: Generalization in 2d and 3d (square and cubic lattice sites) a random walk of N steps in 2d --> a random walk of N/2 steps along the x axis + a random walk of N/2 steps along the y axis for 3d in the same manner:

Suggested further “research” 1. Prove (by computer simulations or analytically if you can…) that 3d 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 1d and 2d? (use square and cubic lattice sites in 2d and 3d, 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 1d, 2d and 3d? 3. Study by computer simulations the  coefficient of a 2d 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)