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Covering Distributions

Alberto M. Campos, Augusto Teixeira

Abstract

In this article, we study a covering process of the discrete one-dimensional torus that uses connected arcs of random sizes in the covering. More precisely, fix a distribution μon \mathbb{N}, and for every n\geq 1 we will cover the torus \mathbb{Z}/n\mathbb{Z} as follows: at each time step, we place an arc with a length distributed as μand a uniform starting point. Eventually, the space will be covered entirely by these arcs. Changing the arc length distribution μcan potentially change the limiting behavior of the covering time. Here, we expose four distinct phases for the fluctuations of the cover time in the limit. These phases can be informally described as the Gumbel phase, the compactly support phase, the pre-exponential phase, and the exponential phase. Furthermore, we expose a continuous-time cover process that works as a limit distribution within the compactly support phase.

Covering Distributions

Abstract

In this article, we study a covering process of the discrete one-dimensional torus that uses connected arcs of random sizes in the covering. More precisely, fix a distribution μon \mathbb{N}, and for every n\geq 1 we will cover the torus \mathbb{Z}/n\mathbb{Z} as follows: at each time step, we place an arc with a length distributed as μand a uniform starting point. Eventually, the space will be covered entirely by these arcs. Changing the arc length distribution μcan potentially change the limiting behavior of the covering time. Here, we expose four distinct phases for the fluctuations of the cover time in the limit. These phases can be informally described as the Gumbel phase, the compactly support phase, the pre-exponential phase, and the exponential phase. Furthermore, we expose a continuous-time cover process that works as a limit distribution within the compactly support phase.
Paper Structure (23 sections, 38 theorems, 221 equations, 7 figures)

This paper contains 23 sections, 38 theorems, 221 equations, 7 figures.

Table of Contents

  1. Introduction and motivation
  2. Gumbel Phase
  3. Proof of Theorem \ref{['teo:1']}
  4. Proof of Propositions \ref{['Prop1']}, \ref{['Prop2']} and \ref{['Prop3']}
  5. Proof of Proposition \ref{['Prop1']}
  6. Proof of Proposition \ref{['Prop2']}
  7. Proof of Proposition \ref{['Prop3']}
  8. Compact Support Phase
  9. Compact support
  10. Non degenerate distribution
  11. Pre-exponential Phase
  12. Exponential Phase
  13. Proof of Theorem B*
  14. The Mandelbrot-Shepp model
  15. Proof of Theorem B*
  16. ...and 8 more sections

Key Result

Theorem A

Assume that $\mathbb{E}\left(R^{1+\varepsilon}\right)<\infty$ for some $\varepsilon>0$, and set $\mu=\mathbb{E}\left(R\right)$. Then, as $n$ goes to infinity: where $\mathbb{P}\left(\mathrm{Gumbel}(0,1)<t\right)=\exp\{-\exp\{-t\}\}$ is the Gumbel distribution with parameters $0$ and $1$.

Figures (7)

  • Figure 1: A representation of the Theorems \ref{['teo:1']} to \ref{['teo:4']} in a line, with the radius distributions $f(r)=\mathbb{P}\left(R\geq r\right)$ disposed in a monotone way. Here $\varepsilon>0$ is a positive small number, and $c>0$ is any constant.
  • Figure 2: A representation side by side of the tree $\mathcal{T}_4$ and the set of intervals $R(i,k)$, on the rectangle $(0,1]\times (0,\infty)$.
  • Figure 3: The black graph is a representative drawn of the sub graph $\hat{\mathcal{T}}$ inside the graph $\mathcal{T}_{{4}}$ drawn as white gray.
  • Figure 4: A representation of the regions $\hat{R}(i,h)$ and intervals $\hat{I}(i,h)$.
  • Figure 5: In the figure the regions corresponding respectively to the objects in the process $W$ and $X$ are drawn using gray and a pattern of lines respectively. Each square region for $W$ corresponds to an object, and each lozangular region for $X$ corresponds with another object. Notice that given a realization of $\omega[1/n]$, in the coupling looking to the set $P_n=\left\{\frac{\ell}{n}: \ell \in\{0,1,\cdots,n\right\}$, every point covered by $W^n$ is also covered by $X^n$.
  • ...and 2 more figures

Theorems & Definitions (70)

  • Theorem A: Gumbel Phase
  • Remark 1
  • Theorem B: Compact Support Phase
  • Theorem B*
  • Theorem C: Pre-Exponential Phase
  • Remark 2
  • Theorem D: Exponential Phase
  • Remark 3
  • Remark 4
  • Lemma 1
  • ...and 60 more