Virtual walks in spin space: a study in a family of two-parameter models

We investigate the dynamics of classical spins mapped as walkers in a virtual "spin" space using a generalised two-parameter family of spin models characterized by parameters $y$ and $z$ [M. J. de Oliveira, J. F. F. Mendes and M. A. Santos, J. Phys. A Math. Gen. \textbf{26}, 2317 (1993)]. The behavior of $S(x,t)$, the probability that the walker is at position $x$ at time $t$ is studied in detail. In general $S(x,t) \sim t^{-\alpha}f(x/t^{\alpha})$ with $\alpha \simeq 1$ or $0.5$ at large times depending on the parameters. In particular, $S(x,t)$ for the point $y=1, z=0.5$ corresponding to the voter model shows a crossover in time; associated with this crossover, two timescales can be defined which vary with the system size $L$ as $L^2\log L$. We also show that as the voter model point is approached from the disordered regions along different directions, the width of the Gaussian distribution $S(x,t)$ diverges in a power law manner with different exponents. For the majority voter case, the results indicate that the the virtual walk can detect the phase transition perhaps more efficiently compared to other non-equilibrium methods.