The Assouad spectrum of random self-affine carpets

TL;DR

This paper investigates the Assouad spectrum and quasi-Assouad dimension of random self-affine carpets, revealing their nuanced behavior and differences from classical dimensions in non-conformal settings.

Contribution

It provides the first almost sure analysis of the Assouad spectrum and quasi-Assouad dimension for random self-affine carpets, highlighting their distinct properties from other fractal dimensions.

Findings

Assouad spectrum and quasi-Assouad dimension often differ from box and Assouad dimensions.

In non-conformal settings, these dimensions do not coincide with classical notions.

The paper introduces new techniques combining deterministic covering with probabilistic estimates.

Abstract

We derive the almost sure Assouad spectrum and quasi-Assouad dimension of random self-affine Bedford-McMullen carpets. Previous work has revealed that the (related) Assouad dimension is not sufficiently sensitive to distinguish between subtle changes in the random model, since it tends to be almost surely `as large as possible' (a deterministic quantity). This has been verified in conformal and non-conformal settings. In the conformal setting, the Assouad spectrum and quasi-Assouad dimension behave rather differently, tending to almost surely coincide with the upper box dimension. Here we investigate the non-conformal setting and find that the Assouad spectrum and quasi-Assouad dimension generally do not coincide with the box dimension or Assouad dimension. We provide examples highlighting the subtle differences between these notions. Our proofs combine deterministic covering techniques with suitably adapted Chernoff estimates and Borel-Cantelli type arguments.