### Abstract

In this paper we consider a cutting process for random mappings. Specifically, for 0 < m < n, we consider the initial (uniform) random mapping digraph G_{n} on n labeled vertices, and we delete (if possible), uniformly and at random, m noncyclic directed edges from G _{n}. The maximal random digraph consisting of the unicyclic components obtained after cutting the m edges is called the trimmed random mapping and is denoted by G_{n}^{m}. If the number of noncyclic directed edges is less than m, then G_{n}^{m} consists of the cycles, including loops, of the initial mapping G_{n}. We consider the component structure of the trimmed mapping G_{n}^{m}. In particular, using the exact distribution we determine the asymptotic distribution of the size of a typical random connected component of G_{n}^{m} as n, m ? 8. This asymptotic distribution depends on the relationship between n and m and we show that there are three distinct cases: (i) m = o(vn), (ii) m = ß vn, where ß > 0 is a fixed parameter, and (iii) vn = o(m). This allows us to study the joint distribution of the order statistics of the normalized component sizes of G_{n}^{m}. When m = o(vn), we obtain the Poisson-Dirichlet(1/2) distribution in the limit, whereas when vn = o(m) the limiting distribution is Poisson-Dirichlet(1). Convergence to the Poisson-Dirichlet(?) distribution breaks down when m = O(vn), and in particular, there is no smooth transition from the PD(1/2) distribution to the PD(1) via the Poisson-Dirichlet distribution as the number of edges cut increases relative to n, the number of vertices in G_{n}. © 2006 Wiley Periodicals, Inc.

Original language | English |
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Pages (from-to) | 287-306 |

Number of pages | 20 |

Journal | Random Structures and Algorithms |

Volume | 30 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - Jan 2007 |

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### Keywords

- Component structure
- Poisson-Dirichlet distribution
- Random mappings

### Cite this

*Random Structures and Algorithms*,

*30*(1-2), 287-306. https://doi.org/10.1002/rsa.20159

}

*Random Structures and Algorithms*, vol. 30, no. 1-2, pp. 287-306. https://doi.org/10.1002/rsa.20159

**A cutting process for random mappings.** / Hansen, Jennie C.; Jaworski, Jerzy.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A cutting process for random mappings

AU - Hansen, Jennie C.

AU - Jaworski, Jerzy

PY - 2007/1

Y1 - 2007/1

N2 - In this paper we consider a cutting process for random mappings. Specifically, for 0 < m < n, we consider the initial (uniform) random mapping digraph Gn on n labeled vertices, and we delete (if possible), uniformly and at random, m noncyclic directed edges from G n. The maximal random digraph consisting of the unicyclic components obtained after cutting the m edges is called the trimmed random mapping and is denoted by Gnm. If the number of noncyclic directed edges is less than m, then Gnm consists of the cycles, including loops, of the initial mapping Gn. We consider the component structure of the trimmed mapping Gnm. In particular, using the exact distribution we determine the asymptotic distribution of the size of a typical random connected component of Gnm as n, m ? 8. This asymptotic distribution depends on the relationship between n and m and we show that there are three distinct cases: (i) m = o(vn), (ii) m = ß vn, where ß > 0 is a fixed parameter, and (iii) vn = o(m). This allows us to study the joint distribution of the order statistics of the normalized component sizes of Gnm. When m = o(vn), we obtain the Poisson-Dirichlet(1/2) distribution in the limit, whereas when vn = o(m) the limiting distribution is Poisson-Dirichlet(1). Convergence to the Poisson-Dirichlet(?) distribution breaks down when m = O(vn), and in particular, there is no smooth transition from the PD(1/2) distribution to the PD(1) via the Poisson-Dirichlet distribution as the number of edges cut increases relative to n, the number of vertices in Gn. © 2006 Wiley Periodicals, Inc.

AB - In this paper we consider a cutting process for random mappings. Specifically, for 0 < m < n, we consider the initial (uniform) random mapping digraph Gn on n labeled vertices, and we delete (if possible), uniformly and at random, m noncyclic directed edges from G n. The maximal random digraph consisting of the unicyclic components obtained after cutting the m edges is called the trimmed random mapping and is denoted by Gnm. If the number of noncyclic directed edges is less than m, then Gnm consists of the cycles, including loops, of the initial mapping Gn. We consider the component structure of the trimmed mapping Gnm. In particular, using the exact distribution we determine the asymptotic distribution of the size of a typical random connected component of Gnm as n, m ? 8. This asymptotic distribution depends on the relationship between n and m and we show that there are three distinct cases: (i) m = o(vn), (ii) m = ß vn, where ß > 0 is a fixed parameter, and (iii) vn = o(m). This allows us to study the joint distribution of the order statistics of the normalized component sizes of Gnm. When m = o(vn), we obtain the Poisson-Dirichlet(1/2) distribution in the limit, whereas when vn = o(m) the limiting distribution is Poisson-Dirichlet(1). Convergence to the Poisson-Dirichlet(?) distribution breaks down when m = O(vn), and in particular, there is no smooth transition from the PD(1/2) distribution to the PD(1) via the Poisson-Dirichlet distribution as the number of edges cut increases relative to n, the number of vertices in Gn. © 2006 Wiley Periodicals, Inc.

KW - Component structure

KW - Poisson-Dirichlet distribution

KW - Random mappings

UR - http://www.scopus.com/inward/record.url?scp=33846671561&partnerID=8YFLogxK

U2 - 10.1002/rsa.20159

DO - 10.1002/rsa.20159

M3 - Article

VL - 30

SP - 287

EP - 306

JO - Random Structures and Algorithms

JF - Random Structures and Algorithms

SN - 1042-9832

IS - 1-2

ER -