Ageing in the exact correlations of the voter model on a fractal
Abstract
The exact behaviour of the enveloppes of the single-time and two-time correlators is found for the voter model on a fractal substrate, with nearest-neighbour interactions. Herein the geometry of the fractal substrate is described by its non-integer geometric fractal dimension d f , and its topology and diffusive transport by the distinct spectral dimension d s . On the level of the equations of motion of the correlators this can be modelled by considering a space-dependent diffusion constant D(r) ∼ r -θ which implies the spectral index θ, itself a function of d f and d s . With a scaling ansatz, the generic phenomenology of ageing is confirmed and the dynamic exponent z = 2 + θ and the autocorrelation exponent λ = d f are derived. The explicitly found dynamic scaling functions are shown to depend only on the spectral dimension d s . The decay of the enveloppe of the density of active interfaces with time is described by the exponent α = 1 -d s /2 for d s < 2, confirming the results of preexisting numerical simulations.
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BibTeX
@misc{henkel2026ageing,
title = {{Ageing in the exact correlations of the voter model on a fractal}},
author = {Malte Henkel},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24408},
eprint = {2609.24408},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05757079v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Ageing in the exact correlations of the voter model on a fractal
AU - Malte Henkel
PY - 2026
DA - 2026/09/21
PB - HAL
DO - 10.48550/arxiv.2609.24408
UR - https://hal.science/hal-05757079v1
AB - The exact behaviour of the enveloppes of the single-time and two-time correlators is found for the voter model on a fractal substrate, with nearest-neighbour interactions. Herein the geometry of the fractal substrate is described by its non-integer geometric fractal dimension d f , and its topology and diffusive transport by the distinct spectral dimension d s . On the level of the equations of motion of the correlators this can be modelled by considering a space-dependent diffusion constant D(r) ∼ r -θ which implies the spectral index θ, itself a function of d f and d s . With a scaling ansatz, the generic phenomenology of ageing is confirmed and the dynamic exponent z = 2 + θ and the autocorrelation exponent λ = d f are derived. The explicitly found dynamic scaling functions are shown to depend only on the spectral dimension d s . The decay of the enveloppe of the density of active interfaces with time is described by the exponent α = 1 -d s /2 for d s < 2, confirming the results of preexisting numerical simulations.
ER -