Percolation of Linked Circles
Let P δ be a Poission process with density,δ in R.3, and let each point in the Poisson process be the center of a unit circle with normal vector a uniform random variable on the 2-sphere. Is there a critical density of the Poisson process such that there exists an infinite unsplittable link of circles? Clearly, if δ=0, then no such infinite unsplittable link exists, and if δ approaches ∞, then it becomes likely that an infinite unsplittable link exists. We prove that there exists an infinite unsplittable link at a finite value of δ. Similar results are true for other models of random closed curves that do not have the volume exclusion property.
KeywordsKnots Links Knotted Polygons Percolation.
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