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 ${\cal D}(r)\sim r^{-θ}$ 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.