# Brownian motion on stable looptrees

## TL;DR

This paper introduces Brownian motion on stable looptrees, establishing its properties through resistance techniques, and analyzes its heat kernel and volume growth, revealing both similarities and differences with stable trees.

## Contribution

It defines Brownian motion on stable looptrees via resistance methods and characterizes it as a scaling limit of discrete random walks, with detailed heat kernel and volume fluctuation analysis.

## Key findings

- Brownian motion on stable looptrees is the scaling limit of random walks on discrete looptrees.
- The heat kernel exhibits precise local and global bounds.
- Volume fluctuations are log-logarithmic locally and logarithmic globally, similar to the Brownian continuum random tree.

## Abstract

In this article, we introduce Brownian motion on stable looptrees using resistance techniques. We prove an invariance principle characterising it as the scaling limit of random walks on discrete looptrees, and prove precise local and global bounds on its heat kernel. We also conduct a detailed investigation of the volume growth properties of stable looptrees, and show that the random volume and heat kernel fluctuations are locally log-logarithmic, and globally logarithmic around leading terms of $r^{\alpha}$ and $t^{\frac{-\alpha}{\alpha + 1}}$ respectively. These volume fluctuations are the same order as for the Brownian continuum random tree, but the upper volume fluctuations (and corresponding lower heat kernel fluctuations) are different to those of stable trees.