Time-Delay Induced Dimensional Crossover in the Voter Model

We investigate the ordering dynamics of the voter model with time-delayed interactions. The dynamical process in the d-dimensional lattice is shown to be equivalent to the first passage problem of a random walker in the (d1)-dimensional strip of a finite width determined by the delay time. The equivalence reveals that the time delay leads to the dimensional crossover from the (d1)-dimensional scaling behavior at a short time to the d-dimensional scaling behavior at a long time. The scaling property in both regimes and the crossover time scale are obtained analytically, which are confirmed with the numerical simulation results.