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### Spatial random permutations with small cycle weights

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Betz, Volker and Ueltschi, Daniel, 1969-.
(2011)
*Spatial random permutations with small cycle weights.*
Probability Theory and Related Fields, Volume 149
(Numbers 1-2).
pp. 191-222.
ISSN 0178-8051

**Full text not available from this repository.**

Official URL: http://dx.doi.org/10.1007/s00440-009-0248-0

## Abstract

We consider the distribution of cycles in two models of random permutations, that are related to one another. In the first model, cycles receive a weight that depends on their length. The second model deals with permutations of points in the space and there is an additional weight that involves the length of permutation jumps. We prove the occurrence of infinite macroscopic cycles above a certain critical density.

Item Type: | Journal Article |
---|---|

Subjects: | Q Science > QA Mathematics |

Divisions: | Faculty of Science > Mathematics |

Library of Congress Subject Headings (LCSH): | Permutations |

Journal or Publication Title: | Probability Theory and Related Fields |

Publisher: | Springer |

ISSN: | 0178-8051 |

Date: | February 2011 |

Volume: | Volume 149 |

Number: | Numbers 1-2 |

Page Range: | pp. 191-222 |

Identification Number: | 10.1007/s00440-009-0248-0 |

Status: | Peer Reviewed |

Publication Status: | Published |

Access rights to Published version: | Restricted or Subscription Access |

Funder: | Engineering and Physical Sciences Research Council (EPSRC), National Science Foundation (U.S.) (NSF) |

Grant number: | EP/D07181X/1 (EPSRC), DMS-0601075 (NSF) |

References: | 1. Baik, J., Deift, P., Johannson, K.: On the distribution of the length of the longest increasing subsequence of random permutations. J. Am. Math. Soc. 12, 1119–1178 (1999) 2. Betz, V., Ueltschi, D.: Spatial random permutations and infinite cycles. Commun. Math. Phys. 285, 469–501 (2009) 3. Betz,V.,Ueltschi, D.,Velenik,Y.: Random permutations with cycle weights. http://arxiv.org/abs/0908. 2217 (2009) 4. Buffet, E., Pulé, J.V.: Fluctuation properties of the imperfect Bose gas. J. Math. Phys. 24, 1608–1616 (1983) 5. Ewens, W.J.: The sampling theory of selectively neutral alleles. Theor. Popul. Biol. 3, 87–112 (1972) 6. Feng, S., Hoppe, F.M.: Large deviation principles for some random combinatorial structures in population genetics and Brownian motion. Ann. Appl. Probab. 8, 975–994 (1998) 7. Ferrari, P., Prähofer, M., Spohn, H.: Stochastic growth in one dimension and Gaussian multi-matrix models. In: XIVth InternationalCongress onMathematical Physics.World Scientific, Singapore (2005) 8. Feynman, R.P.: Atomic theory of the λ transition in Helium. Phys. Rev. 91, 1291–1301 (1953) 9. Gandolfo, D., Ruiz, J., Ueltschi, D.: On a model of random cycles. J. Stat. Phys. 129, 663–676 (2007) 10. Okounkov, A.: The uses of random partitions. In: XIVth International Congress on Mathematical Physics, pp. 379–403. World Scientific, Singapore (2005) 11. Pitman, J.: Exchangeable and partially exchangeable random partitions. Probab. Theory Rel. Fields 102, 145–158 (1995) 12. Ruelle, D.: Statistical Mechanics: Rigorous Results. World Scientific, Singapore (1999) 13. Schramm, O.: Compositions of random transpositions. http://arxiv.org/abs/math/0404356v3 (2004) 14. Shepp, L.A., Lloyd, S.P.: Ordered cycle lengths in a random permutation. Trans. Am. Math. Soc. 121(2), 340–357 (1966) 15. Süt˝o, A.: Percolation transition in the Bose gas. J. Phys. A 26, 4689–4710 (1993) 16. Süt˝o, A.: Percolation transition in the Bose gas II. J. Phys. A 35, 6995–7002 (2002) |

URI: | http://wrap.warwick.ac.uk/id/eprint/41514 |

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