Random Walk with Shrinking Steps

Details Random Walk with Shrinking Steps Random Walk with Shrinking Steps

2003-04-09

arxiv.org/abs/physics/0304036v2

Am. J. Phys. 72, 591-598 (2004)

Physics

Physics Education

9 pages, 5 figures, 2-column revtex4 format. For submission to American Journal of Physics. The 2nd version contains various r

Scientific paper

10.1119/1.1632487

We outline basic properties of a symmetric random walk in one dimension, in which the length of the nth step equals lambda^n, with lambda<1. As the number of steps N-->oo, the probability that the endpoint is at x, P_{lambda}(x;N), approaches a limiting distribution P_{lambda}(x) that has many beautiful features. For lambda<1/2, the support of P_{lambda}(x) is a Cantor set. For 1/2<=lambda<1, there is a countably infinite set of lambda values for which P_{lambda}(x) is singular, while P_{lambda}(x) is smooth for almost all other lambda values. In the most interesting case of lambda=(sqrt{5}-1)/2=g, P_g(x) is riddled with singularities and is strikingly self-similar. The self-similarity is exploited to derive a simple form for the probability measure M(a,b)= int_a^b P_g(x) dx.

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