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We investigate the dynamics of classical spins mapped as walkers in a virtual"spin" space using a generalised two-parameter family of spin modelscharacterized by parameters $y$ and $z$ [M. J. de Oliveira, J. F. F. Mendes andM. 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 $t$ isstudied 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. Inparticular, $S(x,t)$ for the point $y=1, z=0.5$ corresponding to the votermodel shows a crossover in time; associated with this crossover, two timescalescan be defined which vary with the system size $L$ as $L^2\log L$. We also showthat as the voter model point is approached from the disordered regions alongdifferent directions, the width of the Gaussian distribution $S(x,t)$ divergesin a power law manner with different exponents. For the majority voter case,the results indicate that the the virtual walk can detect the phase transitionperhaps more efficiently compared to other non-equilibrium methods.

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