## Abstract

We introduce generalizations of Aldous's Brownian continuous random tree as scaling limits for multicritical models of discrete trees. These discrete models involve trees with fine-tuned vertex-dependent weights ensuring a kth root singularity in their generating function. The scaling limit involves continuous trees with branching points of order up to k+1. We derive explicit integral representations for the average profile of this kth order multicritical continuous random tree, as well as for its history distributions measuring multi-point correlations. The latter distributions involve non-positive universal weights at the branching points together with fractional derivative couplings. We prove universality by rederiving the same results within a purely continuous axiomatic approach based on the resolution of a set of consistency relations for the multi-point correlations. The average profile is shown to obey a fractional differential equation whose solution involves hypergeometric functions and matches the integral formula of the discrete approach.

Original language | English (US) |
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Article number | P04004 |

Journal | Journal of Statistical Mechanics: Theory and Experiment |

Issue number | 4 |

DOIs | |

State | Published - Apr 1 2006 |

Externally published | Yes |

## Keywords

- Correlation functions
- Networks
- Random graphs
- Rigorous results in statistical mechanics

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Statistics and Probability
- Statistics, Probability and Uncertainty