Voter model under stochastic resetting

TL;DR

This paper studies the voter model with stochastic resetting, revealing how resetting influences consensus formation, steady states, and domain wall density across different dimensions, with new kinetic equations and explicit steady-state expressions.

Contribution

It introduces a stochastic resetting mechanism into the voter model, deriving kinetic equations and explicit formulas for steady states and domain wall density.

Findings

Resetting induces a non-equilibrium steady state.

Domain wall density depends on the resetting rate and dimension.

Differentiability at zero resetting rate occurs only for dimensions ≥ 5.

Abstract

The voter model is a toy model of consensus formation based on nearest-neighbor interactions. A voter sits at each vertex in a hypercubic lattice (of dimension $d$) and is in one of two possible opinion states. The opinion state of each voter flips randomly, at a rate proportional to the fraction of the nearest neighbors that disagree with the voter. If the voters are initially independent and undecided, the model is known to lead to a consensus if and only if $d\leq 2$. In this paper the model is subjected to stochastic resetting: the voters revert independently to their initial opinion according to a Poisson process of fixed intensity (the resetting rate). This resetting prescription induces kinetic equations for the average opinion state and for the two-point function of the model. For initial conditions consisting of undecided voters except for one decided voter at the origin, the one-point function evolves as the probability of presence of a diffusive random walker on the lattice, whose position is stochastically reset to the origin. The resetting prescription leads to a non-equilibrium steady state. For an initial state consisting of independent undecided voters, the density of domain walls in the steady state is expressed in closed form as a function of the resetting rate. This function is differentiable at zero if and only if $d\geq 5$.