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We investigate the ordering dynamics of the voter model with time-delayedinteractions. The dynamical process in the $d$-dimensional lattice is shown tobe equivalent to the first passage problem of a random walker in the$(d+1)$-dimensional strip of a finite width determined by the delay time. Theequivalence reveals that the time delay leads to the dimensional crossover fromthe $(d+1)$-dimensional scaling behavior at short time to the $d$-dimensionalscaling behavior in the long time. The scaling property in both regimes and thecrossover time scale are obtained analytically, which are confirmed with thenumerical simulation results.

Time-delay Induced Dimensional Crossover in Voter Model