Multicritical continuous random trees

J. Bouttier, P. Di Francesco, E. Guitter

Research output: Contribution to journalArticlepeer-review


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 languageEnglish (US)
Article numberP04004
JournalJournal of Statistical Mechanics: Theory and Experiment
Issue number4
StatePublished - Apr 1 2006
Externally publishedYes


  • 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


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