# Assouad spectrum thresholds for some random constructions

## TL;DR

This paper introduces a generalized Assouad spectrum for random fractal sets, identifying phase transition thresholds between quasi-Assouad and Assouad dimensions, with explicit results for various models including Galton-Watson trees.

## Contribution

It defines a new interpolating Assouad spectrum and determines phase transition thresholds for several random fractal models, extending understanding of their extremal scaling properties.

## Key findings

- Identified threshold functions for phase transitions in random fractal models.
- Computed thresholds for Galton-Watson tree boundaries.
- Connected stochastic self-similar models to Galton-Watson results.

## Abstract

The Assouad dimension of a metric space determines its extremal scaling properties. The derived notion of the Assouad spectrum fixes relative scales by a scaling function to obtain interpolation behaviour between the quasi-Assouad and box-counting dimensions. While the quasi-Assouad and Assouad dimensions often coincide, they generally differ in random constructions. In this paper we consider a generalised Assouad spectrum that interpolates between the quasi-Assouad to the Assouad dimension. For common models of random fractal sets we obtain a dichotomy of its behaviour by finding a threshold function where the quasi-Assouad behaviour transitions to the Assouad dimension. This threshold can be considered a phase transition and we compute the threshold for the Gromov boundary of Galton-Watson trees and one-variable random self-similar and self-affine constructions. We describe how the stochastically self-similar model can be derived from the Galton-Watson tree result.