Looptree, Fennec, and Snake of ICRT

TL;DR

This paper develops a new theoretical framework for plane inhomogeneous continuum random trees (ICRT) and their associated structures, including looptrees, Gaussian free fields (fennec), and snakes, establishing their properties and fractal dimensions.

Contribution

It introduces a novel theory of plane ICRT and defines associated structures like looptrees, fennec, and snake, proving their continuity, compactness, and fractal properties.

Findings

Looptree is almost surely compact.

Fennec and snake are almost surely continuous.

Fractal dimensions and Hölder exponents are computed.

Abstract

We introduce a new theory of plane $\mathbb R$-tree, to define plane ICRT (inhomogeneous continuum random tree), and its looptree, fennec (a Gaussian free field on the looptree), and snake. We prove that a.s. the looptree is compact, and that a.s. the fennec and snake are continuous. We compute the looptree's fractal dimensions, and the fennec and snake's H\"older exponent. Alongside, we define a Gaussian free field on the ICRT, and prove a condition for its continuity. In a companion paper , we prove that the looptrees, fennecs, and snakes of trees with fixed degree sequence converge toward the looptrees, fennecs and snakes of ICRT.