# Infinite stable looptrees

## TL;DR

This paper constructs an infinite stable looptree, proves its emergence as a limit of discrete and compact looptrees, and analyzes its geometric and stochastic properties, including volume growth and Brownian motion behavior.

## Contribution

It introduces the infinite stable looptree and establishes its role as a limit object in various discrete and compact settings, along with spectral and stochastic analysis.

## Key findings

- Infinite stable looptree constructed and characterized.
- Proven as a local and scaling limit of discrete and compact looptrees.
- Analyzed spectral dimension and Brownian motion on the infinite looptree.

## Abstract

We give a construction of an infinite stable looptree, which we denote by $\mathcal{L}_{\alpha}^{\infty}$, and prove that it arises both as a local limit of the compact stable looptrees of Curien and Kortchemski (2015), and as a scaling limit of the infinite discrete looptrees of Richier (2017) and Bj\"ornberg and Stef\'ansson (2015). As a consequence, we are able to prove various convergence results for volumes of small balls in compact stable looptrees, explored more deeply in a companion paper. We also establish the spectral dimension of $\mathcal{L}_{\alpha}^{\infty}$, and show that it agrees with that of its discrete counterpart. Moreover, we show that Brownian motion on $\mathcal{L}_{\alpha}^{\infty}$ arises as a scaling limit of random walks on discrete looptrees, and as a local limit of Brownian motion on compact stable looptrees, which has similar consequences for the limit of the heat kernel.