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Register a new userThe Zealot Voter Model
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Inspired by the spread of discontent as in the 2016 presidential election, we consider a voter model in which 0s are ordinary voters and 1s are zealots. Thinking of a social network, but desiring the simplicity of an infinite object that can have a nontrivial stationary distribution, space is represented by a tree. The dynamics are a variant of the biased voter: if $x$ has degree $d(x)$ then at rate $d(x)p_k$ the individual at $x$ consults $kge 1$ neighbors. If at least one neighbor is 1, they adopt state 1, otherwise they become 0. In addition at rate $p_0$ individuals with opinion 1 change to 0. As in the contact process on trees, we are interested in determining when the zealots survive and when they will survive locally.
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Given a continuous time Markov Chain ${q(x,y)}$ on a finite set $S$, the associated noisy voter model is the continuous time Markov chain on ${0,1}^S$, which evolves in the following way: (1) for each two sites $x$ and $y$ in $S$, the state at site $x$ changes to the value of the state at site $y$ at rate $q(x,y)$; (2) each site rerandomizes its state at rate 1. We show that if there is a uniform bound on the rates ${q(x,y)}$ and the corresponding stationary distributions are almost uniform, then the mixing time has a sharp cutoff at time $log|S|/2$ with a window of order 1. Lubetzky and Sly proved cutoff with a window of order 1 for the stochastic Ising model on toroids; we obtain the special case of their result for the cycle as a consequence of our result. Finally, we consider the model on a star and demonstrate the surprising phenomenon that the time it takes for the chain started at all ones to become close in total variation to the chain started at all zeros is of smaller order than the mixing time.
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 $q<1$ and clustering 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 for any $m<infty$ the process on the three-dimensional torus with $n$ points survives for time $n^m$, and after an initial transient phase has a density that it is always close to 1/2. 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 $(1/3)log n$.
We study the scaling limit of a large class of voter model perturbations in one dimension, including stochastic Potts models, to a universal limiting object, the continuum voter model perturbation. The perturbations can be described in terms of bulk and boundary nucleations of new colors (opinions). The discrete and continuum (space) models are obtained from their respective duals, the discrete net with killing and Brownian net with killing. These determine the color genealogy by means of reduced graphs. We focus our attention on models where the voter and boundary nucleation dynamics depend only on the colors of nearest neighbor sites, for which convergence of the discrete net with killing to its continuum analog was proved in an earlier paper by the authors. We use some detailed properties of the Brownian net with killing to prove voter model perturbations convergence to its continuum counterpart. A crucial property of reduced graphs is that even in the continuum, they are finite almost surely. An important issue is how vertices of the continuum reduced graphs are strongly approximated by their discrete analogues.
We consider particle systems that are perturbations of the voter model and show that when space and time are rescaled the system converges to a solution of a reaction diffusion equation in dimensions $d ge 3$. Combining this result with properties of the PDE, some methods arising from a low density super-Brownian limit theorem, and a block construction, we give general, and often asymptotically sharp, conditions for the existence of non-trivial stationary distributions, and for extinction of one type. As applications, we describe the phase diagrams of three systems when the parameters are close to the voter model: (i) a stochastic spatial Lotka-Volterra model of Neuhauser and Pacala, (ii) a model of the evolution of cooperation of Ohtsuki, Hauert, Lieberman, and Nowak, and (iii) a continuous time version of the non-linear voter model of Molofsky, Durrett, Dushoff, Griffeath, and Levin. The first application confirms a conjecture of Cox and Perkins and the second confirms a conjecture of Ohtsuki et al in the context of certain infinite graphs. An important feature of our general results is that they do not require the process to be attractive.
Consider a system of particles moving independently as Brownian motions until two of them meet, when the colliding pair annihilates instantly. The construction of such a system of annihilating Brownian motions (aBMs) is straightforward as long as we start with a finite number of particles, but is more involved for infinitely many particles. In particular, if we let the set of starting points become increasingly dense in the real line it is not obvious whether the resulting systems of aBMs converge and what the possible limit points (entrance laws) are. In this paper, we show that aBMs arise as the interface model of the continuous-space voter model. This link allows us to provide a full classification of entrance laws for aBMs. We also give some examples showing how different entrance laws can be obtained via finite approximations. Further, we discuss the relation of the continuous-space voter model to the stepping stone and other related models. Finally, we obtain an expression for the $n$-point densities of aBMs starting from an arbitrary entrance law.
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