Voter model under stochastic resetting

Pascal Grange

Published 2022-07-18Version 1

The voter model is a toy model of consensus formation based on nearest-neighbour interactions. Each voter is endowed with a Boolean variable (a binary opinion) that flips randomly at a rate set according to the opinions of the nearest neighbours. In this paper we subject the system to local resetting by considering stubborn voters. In addition to the usual dynamics, voters revert independently to their initial opinion according to a Poisson process of fixed intensity. The resulting kinetic equations for the average magnetization and two-point function of the model are derived and solved in closed form. They are formally identical to the heat equation coupled to a thermostat, whose temperature profile is induced by the initial conditions. In the case of initial conditions consisting of a single decided voter at the origin in a environment full of undecided voters, the average magnetization evolves as the probability density of a random walker with stochastic resetting to the origin. However, the long-time behaviour of the two-point function contains terms terms that cannot be obtained from a renewal argument.