Virtual walks in the Ising model: a different criticality in two dimensions?

The dynamics of the spins in the Ising model are analysed using a virtual walk scenario. The system is quenched from a very high temperature to a lower one using the Glauber scheme. A walk is associated with each spin which evolves according to the current state of the spin. The known critical temperatures can be accurately detected in both one and two dimensions using the properties of the distribution of the displacements of the walkers. In one dimension, exponentially diverging quantities are obtained close to zero temperature. In two dimensions, a second type of walk involving local energies is considered and several quantities are found to exhibit power law behavior near the critical point. The results in two dimensions suggest an unconventional phase transition occurring in terms of the walk dynamics at the known critical temperature.