Components and Cycles of Random Mappings

Abstract

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%.

The Deepest Cycle of a Random Mapping: a Problem Proposed by Steven Finch

“…Let ν n = ν n (T ) denote its length. In [5], Finch suggests to study the asymptotic behavior of ν n as n → ∞. The main goal of this work is to find the asymptotic of the mean and variance of the size of the deepest cycle ν n .…”

“…It can be interpreted as the approximate limiting probability that the the longest cycle and the largest component of G T are disjoint. Finch [5] called the component containing the longest cycle of a random mapping the richest component. In this terminology, the difference 0.075 equals the approximate limiting probability that the richest component is not the largest one.…”

“…where E 1 (s) denotes the exponential integral function introduced in Theorem 1(i). The last equality in (5) follows from the classical identity Let us now state the integral limit theorem for the size of the largest component of a random mapping: [2, Lemma 5.7]:…”