The q-voter model on the torus

In the $q$-voter model, the voter at $x$ changes its opinion at rate $f_x^q$, where $f_x$ is the fraction of neighbors with the opposite opinion. Mean-field calculations suggest that there should be coexistence between opinions if $q 1$. This model has been extensively studied by physicists, but we do not know of any rigorous results. In this paper, we use the machinery of voter model perturbations to show that the conjectured behavior holds for $q$ close to 1. More precisely, we show that if $q 1$, then the process rapidly reaches fixation on one opinion. It is interesting to note that in the second case the limiting ODE (on its sped up time scale) reaches 0 at time $\log n$ but the stochastic process on the same time scale dies out at time $(1/3)\log n$.

[1]  R. Durrett,et al.  The contact process with fast voting , 2014 .

[2]  J. T. Cox,et al.  Rescaled voter models converge to super-Brownian motion , 2000 .

[3]  T. E. Harris.  On a Class of Set-Valued Markov Processes , 1976 .

[4]  Damon Centola.  How Behavior Spreads: The Science of Complex Contagions , 2018, Science.

[5]  R. Durrett,et al.  The Importance of Being Discrete (and Spatial) , 1994 .

[6]  Romualdo Pastor-Satorras,et al.  Mean-Field Analysis of the q-Voter Model on Networks , 2013, 1301.7563.

[7]  J. T. Cox,et al.  Super-Brownian Limits of Voter Model Clusters , 2001 .

[8]  M. A. Muñoz,et al.  Langevin description of critical phenomena with two symmetric absorbing states. , 2004, Physical review letters.

[9]  Yu Xu,et al.  How behavior spreads: the science of complex contagions , 2019, Information, Communication & Society.

[10]  S. Strogatz,et al.  Linguistics: Modelling the dynamics of language death , 2003, Nature.

[11]  Jon Kleinberg,et al.  Differences in the mechanics of information diffusion across topics: idioms, political hashtags, and complex contagion on twitter , 2011, WWW.

[12]  J. Norris,et al.  Differential equation approximations for Markov chains , 2007, 0710.3269.

[13]  Maxi San Miguel,et al.  Fragmentation transitions in a coevolving nonlinear voter model , 2017, Scientific Reports.

[14]  Jari Saramäki,et al.  Dynamics of latent voters. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[15]  R. Durrett,et al.  Latent Voter Model on Locally Tree-Like Random Graphs , 2016, 1605.09039.

[16]  Richard Durrett,et al.  Voter Model Perturbations and Reaction Diffusion Equations , 2011, Astérisque.

[17]  T. Liggett,et al.  Stochastic Interacting Systems: Contact, Voter and Exclusion Processes , 1999 .

[18]  Arkadiusz Jędrzejewski,et al.  Pair approximation for the q-voter model with independence on complex networks. , 2017, Physical review. E.

[19]  Flávio L. Pinheiro,et al.  Consensus and polarization in competing complex contagion processes , 2018, Journal of the Royal Society Interface.

[20]  Iljana Zähle.  Renormalization of the Voter Model in Equilibrium , 2001 .

[21]  Frank E. Grubbs,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[22]  J. T. Cox,et al.  Evolutionary Games on the Torus with Weak Selection , 2015, 1511.04713.

[23]  Maury Bramson,et al.  Asymptotics for interacting particle systems onZd , 1980 .

[24]  Masanobu Taniguchi,et al.  Elements of Stochastic Processes , 2000 .

[25]  Claudia Neuhauser,et al.  Particle Systems and Reaction-Diffusion Equations , 1994 .

[26]  F. Vazquez,et al.  Agent based models of language competition: macroscopic descriptions and order–disorder transitions , 2010, 1002.1251.

[27]  Matthias Abend,et al.  Additive And Cancellative Interacting Particle Systems , 2016 .