Random Walks

The main mathematical news

2006: A formula for the average number of loops around an object in an n-step random walk on a plane.

Moreover, if n tends to infinity, the number of loops tends to infinity as well.

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Additional Theorems / conjectures / Open questions

*   1921: George Polya. An indefinite linear random walk (unit steps forward/backwards by the results of a coin toss) or planar random walk (unit steps forward/backward/left/right by the results of tetrahedral die), will return to the origin with probability 1, more than once

*   A 3D random walk, a return to the origin – even once – is not guaranteed, even if it goes on indefinitely. (In mathematical terms: the probability is less than 1).

*   Does the price of a fluctuating stock vary like a random walk?

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The main mathematical concepts / Principles

Stochastic Processes


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