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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: ---

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

----- 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: ---

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 