Components and Cycles of Random Mappings

05/11/2022

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by   Steven Finch, et al.

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Each connected component of a mapping {1,2,...,n}→{1,2,...,n} contains a unique cycle. The largest such component can be studied probabilistically via either a delay differential equation or an inverse Laplace transform. The longest such cycle likewise admits two approaches: we find an (apparently new) density formula for its length. Implications of a constraint – that exactly one component exists – are also examined. For instance, the mean length of the longest cycle is (0.7824...)√(n) in general, but for the special case, it is (0.7978...)√(n), a difference of less than 2%.