Drunkard's Walk

Simulation of a 2D random walk with 229 steps and a random step size from the interval [−0.5; 0.5] for the x and y directions

The Drunkard's Walk (English for the path of the drunk is) an image from the theory of probability , the movement illustrating a random (random walk, Random Walk ) is used. It was probably coined in 1905 by a letter from Karl Pearson in Nature magazine , inspired by studying the distribution of insect populations.

“A man starts from a point and walks yards in a straight line; he then turns through any angle whatever and walks another yards in a second straight line. He repeats this process times. I require the probability that after these stretches he is at a distance between and from his starting point,. " ${\displaystyle O}$ ${\displaystyle l}$ ${\displaystyle l}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle r}$ ${\displaystyle r+dr}$ ${\displaystyle O}$

“A person starts at a point and walks meters straight ahead; then he turns around any angle and walks straight ahead in the new direction . He repeats this time. I am looking for the probability that after these movements he will be in the distance between and from the starting point . ${\displaystyle O}$ ${\displaystyle l}$ ${\displaystyle l}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle r}$ ${\displaystyle r+dr}$ ${\displaystyle O}$

The image of a drunk who walks a certain distance straight ahead, then loses balance and orientation and walks the same distance in a random different direction fits this sequence of movements.

A far-reaching solution to the original problem was given by John William Strutt, 3rd Baron Rayleigh , in another letter to the editor in the same volume. Rayleigh's solution is that the probability distribution of the random positions of the drunkard is approximated after many steps from the normal distribution .

The term is usually used for random movements according to a similar scheme, the simplest, often considered case is the random walk on the number line. The “drunk” moves with steps of length l , each time randomly to the left with a fixed probability p , and correspondingly to the right with the probability 1− p .

The question of how likely it is that the stranger will return to the origin depends surprisingly on the dimensionality of the space. Shizuo Kakutani put it, staying in the picture:

"A drunk man will find his way home, but a drunk bird may get lost forever."

"A drunk person finds home, but a drunk bird can be lost forever."

This refers to Pólya's theorem of 1921, published in the Mathematische Annalen, in which the recurrence of random walks in the plane was proven, which no longer applies in three-dimensional space.

literature

• Barry D. Hughes: Random Walks and Random Environments: Volume 1: Random Walks. Oxford University Press, USA 1995, ISBN 0-19-853788-3 .

Web links

• Problem in a letter to the editor from Karl Pearson and John William Strutt, 3rd Baron Rayleigh , in Nature 72, 1905, pp. 294, 318, 342 (see individual references)

Individual evidence

1. ^ Karl Pearson: The Problem of the Random Walk . In: Nature . tape 72 , no. 1865 , July 1, 1905, p. 294 , doi : 10.1038 / 072294b0 (English).
2. ^ L. Rayleigh: The problem of the random walk . In: Nature . tape 72 , no. 1866 , 1905, pp. 318 , doi : 10.1038 / 072318a0 (English).
3. ^ Karl Pearson: The Problem of the Random Walk . In: Nature . tape 72 , no. 1867 , 1 August 1905, p 342 , doi : 10.1038 / 072342a0 (English).
4. ^ ^{A b} Reinhard Mahnke, Jevgenijs Kaupužs, Ihor Lubashevsky: Physics of stochastic processes: how randomness acts in time . Wiley-VCH, Weinheim 2008, ISBN 3-527-40840-1 , pp. 181 (English).
5. ↑ Georg Pólya: About a problem of the calculation of probability concerning the random walk in the road network . In: Mathematical Annals . tape 84 , no. 1-2 , March 1921, pp. 149-160 , doi : 10.1007 / BF01458701 .