dorsal/arxiv
View SchemaRandom Walk with Shrinking Steps
| Authors | P. L. Krapivsky, S. Redner |
|---|---|
| Categories | |
| ArXiv ID | physics/0304036 |
| URL | https://arxiv.org/abs/physics/0304036 |
| DOI | 10.1119/1.1632487 |
| Journal | Am. J. Phys. 72, 591-598 (2004) |
Abstract
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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"abstract": "We outline basic properties of a symmetric random walk in one dimension, in\nwhich the length of the nth step equals lambda^n, with lambda\u003c1. As the number\nof steps N--\u003eoo, the probability that the endpoint is at x, P_{lambda}(x;N),\napproaches a limiting distribution P_{lambda}(x) that has many beautiful\nfeatures. For lambda\u003c1/2, the support of P_{lambda}(x) is a Cantor set. For\n1/2\u003c=lambda\u003c1, there is a countably infinite set of lambda values for which\nP_{lambda}(x) is singular, while P_{lambda}(x) is smooth for almost all other\nlambda values. In the most interesting case of lambda=(sqrt{5}-1)/2=g, P_g(x)\nis riddled with singularities and is strikingly self-similar. The\nself-similarity is exploited to derive a simple form for the probability\nmeasure M(a,b)= int_a^b P_g(x) dx.",
"arxiv_id": "physics/0304036",
"authors": [
"P. L. Krapivsky",
"S. Redner"
],
"categories": [
"physics.ed-ph",
"cond-mat.stat-mech",
"math.PR",
"physics.gen-ph"
],
"doi": "10.1119/1.1632487",
"journal_ref": "Am. J. Phys. 72, 591-598 (2004)",
"title": "Random Walk with Shrinking Steps",
"url": "https://arxiv.org/abs/physics/0304036"
},
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