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Random Walk Applet

 1 dimension: The blue point is moving on a line with integer coordinates: The 2 directions of a single step: x+1, x-1 2 dimensions: The blue point is moving in a plane with integer coordinates: The 4 directions of a single step: x+1, x-1, y+1, y-1

 select from the menu button starting a single walk, maximum of n=1000 steps, the diagram at the bottom is showing the distances d(n) button to stop the walk

1 dimension:

An interesting question arising in the study of random walks concerns
the probability of returning to the initial position (origin, "equalization").

The probability P(n) of return to origin at step n (n even) is:

For large n (even):

Graph of the first (strict) formula:

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Applet results:

The total number of returns to origin (within a fixed number n of steps) is proportional
to the number N of walks:

The probalibity for n=100 steps is 0.076

2 dimensions:

Example:

100 steps, final position (11|7),
the distance from origin is d = sqrt(x2+y2) = 13.04

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and for large n:

In 2 dimensions the probability is, of course, the square of
the one in 1 dimension, requiring x=0 AND y=0

Graph of the first (strict) formula:

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Applet results:

The probalibity for n=50 steps is 0.014

Statistical analysis

 button starting a set of N walks the numbers of steps and walks can be selected from the menus

The mean squared distance is proportional to the number n of steps:

 Books Küppers, Bernd-Olaf: Die Berechenbarkeit der Welt, Grenzfragen der exakten Wissenschaften. S. Hirzel, Stuttgart 2012. Entropie und Zeitstruktur, S. 200-210 Eigen, Manfred, and Winkler, Ruth: Das Spiel, Naturgesetze steuern den Zufall. Pieper, München 1975. Kapitel 4:Statistische Kugelspiele Web Links Random Walk--1-Dimensional (Wolfram MathWorld) Random Walk--2-Dimensional (Wolfram MathWorld) A 1D Random Walk Visits The Origin Infinitely Often

Updated: 2012 Sep 22