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On the Deepest Cycle of a Random Mapping

Ljuben Mutafchiev, Steven Finch

Abstract

Let $\mathcal{T}_n$ be the set of all mappings $T:\{1,2,\ldots,n\}\to\{1,2,\ldots,n\}$. The corresponding graph of $T$ is a union of disjoint connected unicyclic components. We assume that each $T\in\mathcal{T}_n$ is chosen uniformly at random (i.e., with probability $n^{-n}$). The cycle of $T$ contained within its largest component is callled the deepest one. For any $T\in\mathcal{T}_n$, let $ν_n=ν_n(T)$ denote the length of this cycle. In this paper, we establish the convergence in distribution of $ν_n/\sqrt{n}$ and find the limits of its expectation and variance as $n\to\infty$. For $n$ large enough, we also show that nearly $55\%$ of all cyclic vertices of a random mapping $T\in\mathcal{T}_n$ lie in the deepest cycle and that a vertex from the longest cycle of $T$ does not belong to its largest component with approximate probability $0.075$.

On the Deepest Cycle of a Random Mapping

Abstract

Let be the set of all mappings . The corresponding graph of is a union of disjoint connected unicyclic components. We assume that each is chosen uniformly at random (i.e., with probability ). The cycle of contained within its largest component is callled the deepest one. For any , let denote the length of this cycle. In this paper, we establish the convergence in distribution of and find the limits of its expectation and variance as . For large enough, we also show that nearly of all cyclic vertices of a random mapping lie in the deepest cycle and that a vertex from the longest cycle of does not belong to its largest component with approximate probability .
Paper Structure (4 sections, 5 theorems, 55 equations)

This paper contains 4 sections, 5 theorems, 55 equations.

Table of Contents

  1. Introduction and Statement of the Main Result
  2. Preliminary Results
  3. Proof of Theorem 1
  4. Concluding Remarks

Key Result

Theorem 1

(i) As $n\to\infty$, $n^{-1/2}\nu_n\to_d\sqrt{\chi^2(1)\mu}$, where $\chi^2(1)$ and $\mu$ are independent random variables with distribution functions $G(x)$ and $F(x)$ given by ( ) and ( ), respectively. (ii) Let $E_1(s)=\int_s^\infty\frac{e^{-t}}{t}dt, s>0,$ be the exponential integral function. Then we have (iii) We also have

Theorems & Definitions (7)

  • Theorem 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • Lemma 4