Multicritical continuous random trees

TL;DR

This paper introduces a family of multicritical continuous random trees as scaling limits of discrete trees with fine-tuned weights, deriving explicit formulas and proving universality through both discrete and continuous methods.

Contribution

It generalizes Aldous' Brownian CRT to multicritical cases, providing explicit integral representations and a unified axiomatic approach for these new continuous tree models.

Findings

Derived explicit integral formulas for average profiles.

Established universality via continuous axiomatic approach.

Connected fractional differential equations with hypergeometric solutions.

Abstract

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.