Diffusion on non exactly decimable tree-like fractals

TL;DR

This paper introduces a new analytical method to calculate the spectral dimension of complex tree-like fractals, revealing their diffusion properties and structural characteristics, with implications for understanding non-integer dimensions and phase transitions.

Contribution

It presents a novel invariance-based analytical technique for spectral dimension calculation on a broad class of fractals, extending previous methods and uncovering unique diffusion behaviors.

Findings

Spectral dimension is non-integer and ≥ 2 for these fractals.

Diffusion laws are non anomalous, indicating normal diffusion behavior.

No phase transitions occur in spin models on these fractals.

Abstract

We calculate the spectral dimension of a wide class of tree-like fractals by solving the random walk problem through a new analytical technique, based on invariance under generalized cutting-decimation transformations. These fractals are generalizations of the NTD lattices and they are characterized by non integer spectral dimension equal or greater then 2, non anomalous diffusion laws, dynamical dimension splitting and absence of phase transitions for spin models.