Time-Delay Induced Dimensional Crossover in the Voter Model

Abstract

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 (d + 1)-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 (d + 1)-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.PACS numbers: 02.70.Rr Interactions among spatially distributed elements are subjected to a time delay. Any physical interaction propagates at a finite speed. Complex interactions among biological oscillators are mediated by biochemical materials moving at a finite speed [1]. Such a limited signal propagation speed causes a time delay. When a delay time is comparable to or larger than characteristic time scales, the time delay brings about rich phenomena. It leads to multistable synchronized states for phase oscillators [1][2][3][4][5][6][7], amplitude death for coupled limitcycle oscillators [8,9], stabilization of periodic orbits in chaotic systems [10], a resonant behavior in stochastic systems [11][12][13], dynamic instability in feedback control systems [14], a pattern formation in evolutionary game dynamics [15,16], and so on.In this Letter, we investigate the effects of time delay on the ordering dynamics in the context of the voter model (VM). The VM is a prototypical model for opinion dynamics [17][18][19][20][21][22]. In the VM, a voter takes one of the two opinions represented by an Ising spin variable. Each voter selects randomly one of its nearest neighbors and updates its spin state by copying that of the selected neighbor. Note that the interaction in the VM is instantaneous. In a realistic situation, however, an information propagates at a finite speed so that a voter can have an access to the past states of others. Thus, it is natural to consider the time-delayed interaction in the opinion dynamics, which has never been studied before. We introduce a time-delayed voter model (DVM) by incorporating the time delay into the VM and investigate the dynamic scaling behavior. We find that the time delay leads to the dimensional crossover: the DVM in d dimension displays the (d + 1)-dimensional scaling behavior at a short time and then d-dimensional scaling behavior at a long time.The time delay makes the dynamics of the DVM nonMarkovian. Recently, the non-Markovian generalizations of the VM have been considered by introducing the latency period, the inertia effect, or non-Poissonian interevent interval distributions [23][24][25][26]. Non-Markovian dynamics of Ising-like systems with time-delayed interactions has also been studied in Refs. [27][28][29]. Our study reveals a new aspect of time-delayed interactions.The DVM in the d-dimensional hypercube Z d is defined as fol...

Non-Markovian Majority-Vote model

“…For example, the presence of non-trivial structure in the interacting patterns such as heavy-tailed degree distribution [4,8], mesoscopic structures [9,10], multilayer structures [11][12][13], can induce significant change in the dynamics. Moreover relevant effect can be obtained also changing the dynamical rules by introducing of more than two states [14,15], time delay [16], nonhomogeneous interevent intervals [17][18][19][20] or a fraction of zealot [21,22].…”