Diffusion and spectral dimension on Eden tree

TL;DR

This paper investigates the spectral properties of random walks on Eden trees in two and three dimensions, revealing finite-size effects and complex long-time behaviors that challenge existing scaling relations.

Contribution

It provides the first detailed calculation of the eigenspectrum and spectral dimension of Eden trees, highlighting finite-size effects and deviations from expected scaling laws.

Findings

Finite-size effects cause crossover behaviors.

Spectral dimension and walk dimension are computed.

Scaling relation violations observed at short times.

Abstract

We calculate the eigenspectrum of random walks on the Eden tree in two and three dimensions. From this, we calculate the spectral dimension $d_s$ and the walk dimension $d_w$ and test the scaling relation $d_s = 2d_f/d_w$ ($=2d/d_w$ for an Eden tree). Finite-size induced crossovers are observed, whereby the system crosses over from a short-time regime where this relation is violated (particularly in two dimensions) to a long-time regime where the behavior appears to be complicated and dependent on dimension even qualitatively.