Physics – Mathematical Physics
Scientific paper
2006-03-02
J. Stat. Mech. (2006) P04004
Physics
Mathematical Physics
34 pages, 12 figures, uses lanlmac, hyperbasics, epsf
Scientific paper
10.1088/1742-5468/2006/04/P04004
We introduce generalizations of Aldous' 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 k-th 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 k-th 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.
Bouttier Jérémie
Francesco Philippe Di
Guitter Emmanuel
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