Clustering and coexistence

The interest in is the limiting behavior of the models. Since the flip rates of a site depends on its neighbours, it is obvious that when all sites take the same value, the whole system stops changing forever. Therefore, a voter model has two trivial extremal stationary distributions, the point-masses ${\displaystyle \scriptstyle \delta _{0}}$ and ${\displaystyle \scriptstyle \delta _{1}}$ on ${\displaystyle \scriptstyle \eta \equiv 0}$ or ${\displaystyle \scriptstyle \eta \equiv 1}$ respectively, which represent consensus. The main question to be discussed is whether or not there are others, which would then represent coexistence of different opinions in equilibrium. It is said coexistence occurs if there is a stationary distribution that concentrates on configurations with infinitely many 0's and 1's. On the other hand, if for all ${\displaystyle \scriptstyle x,y\in Z^{d}}$ and all initial configurations, then

${\displaystyle \lim _{t\rightarrow \infty }P[\eta _{t}(x)\neq \eta _{t}(y)]=0}$

It is said that clustering occurs.

It is important to distinguish clustering with the concept of cluster. Clusters are defined to be the connected components of ${\displaystyle \scriptstyle \{x:\eta (x)=0\}}$ or ${\displaystyle \scriptstyle \{x:\eta (x)=1\}}$.

Model description

This section will be dedicated to one of the basic voter models, the Linear Voter Model.

If ${\displaystyle \scriptstyle p(}$•,•${\displaystyle \scriptstyle )}$ be the transition probabilities for an irreducible random walk on ${\displaystyle \scriptstyle Z^{d}}$, then:

${\displaystyle p(x,y)\geq 0\quad {\text{and}}\sum _{y}p(x,y)=1}$

Then in Linear voter model, the transition rates are linear functions of ${\displaystyle \scriptstyle \eta }$:

${\displaystyle c(x,\eta )=\left\{{\begin{array}{l}\sum _{y}p(x,y)\eta (y)\quad {\text{for all}}\quad \eta (x)=0\\\sum _{y}p(x,y)(1-\eta (y))\quad {\text{for all}}\quad \eta (x)=1\\\end{array}}\right.}$

Or if ${\displaystyle \scriptstyle \eta _{x}}$ indicates that a flip happens at ${\displaystyle \scriptstyle x}$, then transition rates are simply:

${\displaystyle \eta \rightarrow \eta _{x}\quad {\text{at rate}}\sum _{y:\eta (y)\neq \eta (x)}p(x,y).}$

A process of coalescing random walks ${\displaystyle \scriptstyle A_{t}\subset Z^{d}}$ is defined as follows. Here ${\displaystyle \scriptstyle A_{t}}$ denotes the set of sites occupied by these random walks at time ${\displaystyle \scriptstyle t}$. To define ${\displaystyle \scriptstyle A_{t}}$, consider several (continuous time) random walks on ${\displaystyle \scriptstyle Z^{d}}$ with unit exponential holding times and transition probabilities ${\displaystyle \scriptstyle p(}$•,•${\displaystyle \scriptstyle )}$, and take them to be independent until two of them meet. At that time, the two that meet coalesce into one particle, which continues to move like a random walk with transition probabilities ${\displaystyle \scriptstyle p(}$•,•${\displaystyle \scriptstyle )}$ .

The concept of Duality is essential for analysing the behavior of the voter models. The linear voter models satisfy a very useful form of duality, known as coalescing duality, which is:

${\displaystyle P^{\eta }(\eta _{t}\equiv 1\quad {\text{on }}A)=P^{A}(\eta (A_{t})\equiv 1),}$

where ${\displaystyle \scriptstyle \eta \in \{0,1\}^{Z^{d}}}$ is the initial configuration of ${\displaystyle \scriptstyle \eta _{t}}$ and ${\displaystyle \scriptstyle A=\{x\in Z^{d},\eta (x)=1\}\subset Z^{d}}$ is the initial state of the coalescing random walks ${\displaystyle \scriptstyle A_{t}}$.

Limiting behaviors of linear voter models

Let ${\displaystyle \scriptstyle p(x,y)}$ be the transition probabilities for an irreducible random walk on ${\displaystyle \scriptstyle Z^{d}}$ and ${\displaystyle \scriptstyle p(x,y)=p(0,x-y)}$, then the duality relation for such linear voter models says that ${\displaystyle \scriptstyle \forall \eta \in S=\{0,1\}^{Z^{d}}}$

${\displaystyle P^{\eta }[\eta _{t}(x)\neq \eta _{t}(y)]=P[\eta (X_{t})\neq \eta (Y_{t})]}$

where ${\displaystyle \scriptstyle X_{t}}$ and ${\displaystyle \scriptstyle Y_{t}}$ are (continuous time) random walks on ${\displaystyle \scriptstyle Z^{d}}$ with ${\displaystyle \scriptstyle X_{0}=x}$, ${\displaystyle \scriptstyle Y_{0}=y}$, and ${\displaystyle \scriptstyle \eta (X_{t})}$ is the position taken by the random walk at time ${\displaystyle \scriptstyle t}$. ${\displaystyle \scriptstyle X_{t}}$ and ${\displaystyle \scriptstyle Y_{t}}$ forms a coalescing random walks described at the end of section 2.1. ${\displaystyle \scriptstyle X(t)-Y(t)}$ is a symmetrized random walk. If ${\displaystyle \scriptstyle X(t)-Y(t)}$ is recurrent and ${\displaystyle \scriptstyle d\leq 2}$, ${\displaystyle \scriptstyle X_{t}}$ and ${\displaystyle \scriptstyle Y_{t}}$ will hit eventually with probability 1, and hence

${\displaystyle P^{\eta }[\eta _{t}(x)\neq \eta _{t}(y)]=P[\eta (X_{t})\neq \eta (Y_{t})]\leq P[X_{t}\neq Y_{t}]\rightarrow 0\quad {\text{as}}\quad t\to 0}$

Therefore, the process clusters.

On the other hand, when ${\displaystyle d\geq 3}$, the system coexists. It is because for ${\displaystyle \scriptstyle d\geq 3}$, ${\displaystyle \scriptstyle X(t)-Y(t)}$ is transient, thus there is a positive probability that the random walks never hit, and hence for ${\displaystyle \scriptstyle x\neq y}$

${\displaystyle \lim _{t\rightarrow \infty }P[\eta _{t}(x)\neq \eta _{t}(y)]=C\lim _{t\rightarrow \infty }P[X_{t}\neq Y_{t}]>0}$

for some constant ${\displaystyle C}$ corresponding to the initial distribution.

If ${\displaystyle \scriptstyle {\tilde {X}}(t)=X(t)-Y(t)}$ be a symmetrized random walk, then there are the following theorems:

Theorem 2.1

The linear voter model ${\displaystyle \scriptstyle \eta _{t}}$ clusters if ${\displaystyle \scriptstyle {\tilde {X}}_{t}}$ is recurrent, and coexists if ${\displaystyle \scriptstyle {\tilde {X}}_{t}}$ is transient. In particular,

1. the process clusters if ${\displaystyle \scriptstyle d=1}$ and ${\displaystyle \scriptstyle \sum _{x}|x|p(0,x)\leq \infty }$, or if ${\displaystyle \scriptstyle d=2}$ and ${\displaystyle \scriptstyle \sum _{x}|x|^{2}p(0,x)\leq \infty }$;
2. the process coexists if ${\displaystyle \scriptstyle d\geq 3}$.

Remarks: To contrast this with the behavior of the threshold voter models that will be discussed in next section, note that whether the linear voter model clusters or coexists depends almost exclusively on the dimension of the set of sites, rather than on the size of the range of interaction.

Theorem 2.2 Suppose ${\displaystyle \scriptstyle \mu }$ is any translation spatially ergodic and invariant probability measure on the state space ${\displaystyle \scriptstyle S=\{0,1\}^{Z^{d}}}$, then

1. If ${\displaystyle \scriptstyle {\tilde {X}}_{t}}$ is recurrent, then ${\displaystyle \scriptstyle \mu S(t)\Rightarrow \rho \delta _{1}+(1-\rho )\delta _{0}\quad {\text{as}}\quad t\to \infty }$;
2. If ${\displaystyle \scriptstyle {\tilde {X}}_{t}}$ is transient, then ${\displaystyle \scriptstyle \mu S(t)\Rightarrow \mu _{\rho }}$.

where ${\displaystyle \scriptstyle \mu S(t)}$ is the distribution of ${\displaystyle \scriptstyle \eta _{t}}$; ${\displaystyle \scriptstyle \Rightarrow }$ means weak convergence, ${\displaystyle \scriptstyle \mu _{\rho }}$ is a nontrivial extremal invariant measure and ${\displaystyle \scriptstyle \rho =\mu (\{\eta$ :\eta (x)=1\})} .

A special linear voter model

One of the interesting special cases of the linear voter model, known as the basic linear voter model, is that for state space ${\displaystyle \scriptstyle \{0,1\}^{Z^{d}}}$:

${\displaystyle p(x,y)={\begin{cases}1/2d&{\text{if }}|x-y|=1{\text{ and }}\eta (x)\neq \eta (y)\\[8pt]0&{\text{otherwise}}\end{cases}}}$

So that

${\displaystyle \eta _{t}(x)\to 1-\eta _{t}(x)\quad {\text{at rate}}\quad (2d)^{-1}|\{y:|y-x|=1,\eta _{t}(y)\neq \eta _{t}(x)\}|}$

In this case, the process clusters if ${\displaystyle \scriptstyle d\leq 2}$, while coexists if ${\displaystyle \scriptstyle d\geq 3}$. This dichotomy is closely related to the fact that simple random walk on ${\displaystyle \scriptstyle Z^{d}}$ is recurrent if ${\displaystyle \scriptstyle d\leq 2}$ and transient if ${\displaystyle \scriptstyle d\geq 3}$.

Model description

This section concentrates on a kind of non-linear voter model, known as the threshold voter model. To define it, let ${\displaystyle \scriptstyle {\mathcal {N}}}$ be a neighbourhood of ${\displaystyle \scriptstyle 0\in Z^{d}}$ that is obtained by intersecting ${\displaystyle \scriptstyle Z^{d}}$ with any compact, convex, symmetric set in ${\displaystyle \scriptstyle R^{d}}$; in other words, ${\displaystyle \scriptstyle {\mathcal {N}}}$ is assumed to be a finite set that is symmetric with respect to all reflections and irreducible (i.e. the group it generates is ${\displaystyle \scriptstyle Z^{d}}$). It can always be assumed that ${\displaystyle \scriptstyle {\mathcal {N}}}$ contains all the unit vectors ${\displaystyle \scriptstyle (1,0,0,\dots ,0),\dots ,(0,\dots ,0,1)}$. For a positive integer ${\displaystyle \scriptstyle T}$, the threshold voter model with neighbourhood ${\displaystyle \scriptstyle {\mathcal {N}}}$ and threshold ${\displaystyle \scriptstyle T}$ is the one with rate function:

${\displaystyle c(x,\eta )=\left\{{\begin{array}{l}1\quad {\text{if}}\quad |\{y\in x+{\mathcal {N}}:\eta (y)\neq \eta (x)\}|\geq T\\0\quad {\text{otherwise}}\\\end{array}}\right.}$

Simply put, the transition rate of site ${\displaystyle \scriptstyle x}$ is 1 if the number of sites that do not take the same value is larger or equal to the threshold T. Otherwise, site ${\displaystyle \scriptstyle x}$ stays at the current status and will not flip.

For example, if ${\displaystyle \scriptstyle d=1}$, ${\displaystyle \scriptstyle {\mathcal {N}}=\{-1,0,1\}}$ and ${\displaystyle \scriptstyle T=2}$, then the configuration ${\displaystyle \scriptstyle \dots 1\quad 1\quad 0\quad 0\quad 1\quad 1\quad 0\quad 0\dots }$ is an absorbing state or a trap for the process.

Limiting behaviors of threshold voter model

If a threshold voter model does not fixate, the process should be expected to will coexist for small threshold and cluster for large threshold, where large and small are interpreted as being relative to the size of the neighbourhood, ${\displaystyle \scriptstyle |{\mathcal {N}}|}$. The intuition is that having a small threshold makes it easy for flips to occur, so it is likely that there will be a lot of both 0's and 1's around at all times. The following are three major results:

1. If ${\displaystyle \scriptstyle T>{\frac {|{\mathcal {N}}|-1}{2}}}$, then the process fixates in the sense that each site flips only finitely often.
2. If ${\displaystyle \scriptstyle d=1}$ and ${\displaystyle \scriptstyle T={\frac {|{\mathcal {N}}|-1}{2}}}$, then the process clusters.
3. If ${\displaystyle \scriptstyle T=\theta |{\mathcal {N}}|}$ with ${\displaystyle \scriptstyle \theta }$ sufficiently small(${\displaystyle \scriptstyle \theta <{\frac {1}{4}}}$) and ${\displaystyle \scriptstyle |{\mathcal {N}}|}$ sufficiently large, then the process coexists.

Here are two theorems corresponding to properties (1) and (2).

Theorem 3.1

If ${\displaystyle \scriptstyle T>{\frac {|{\mathcal {N}}|-1}{2}}}$, then the process fixates.

Theorem 3.2

The threshold voter model in one dimension (${\displaystyle \scriptstyle d=1}$) with ${\displaystyle \scriptstyle {\mathcal {N}}=\{-T,\dots ,T\},T\geq 1}$, clusters.

proof

The idea of the proof is to construct two sequences of random times ${\displaystyle \scriptstyle U_{n}}$, ${\displaystyle \scriptstyle V_{n}}$ for ${\displaystyle \scriptstyle n\geq 1}$ with the following properties:

1. ${\displaystyle \scriptstyle 0=V_{0}<U_{1}<V_{1}<U_{2}<V_{2}<\dots }$,
2. ${\displaystyle \scriptstyle \{U_{k+1}-V_{k},k\geq 0\}}$ are i.i.d.with ${\displaystyle \scriptstyle \mathrm {E} (U_{k+1}-V_{k})<\infty }$,
3. ${\displaystyle \scriptstyle \{V_{k}-U_{k},k\geq 1\}}$ are i.i.d.with ${\displaystyle \scriptstyle \mathrm {E} (V_{k}-U_{k})=\infty }$,
4. the random variables in (b) and (c) are independent of each other,
5. event A=${\displaystyle \scriptstyle \{\eta _{t}(.)}$ is constant on ${\displaystyle \scriptstyle \{-T,\dots ,T\}\}}$, and event A holds for every ${\displaystyle \scriptstyle t\in \cup _{k=1}^{\infty }[U_{k},V_{k}]}$.

Once this construction is made, it will follow from renewal theory that

${\displaystyle P(A)\geq P(t\in \cup _{k=1}^{\infty }[U_{k},V_{k}])\to 1\quad {\text{as}}\quad t\to \infty }$

Hence,${\displaystyle \scriptstyle \lim _{t\rightarrow \infty }P(\eta _{t}(1)\neq \eta _{t}(0))=0}$, so that the process clusters.

Remarks: (a) Threshold models in higher dimensions do not necessarily cluster if ${\displaystyle \scriptstyle T={\frac {|{\mathcal {N}}|-1}{2}}}$. For example, take ${\displaystyle \scriptstyle d=2,T=2}$ and ${\displaystyle \scriptstyle {\mathcal {N}}=\{(0,0),(0,1),(1,0),(0,-1),(-1,0)\}}$. If ${\displaystyle \scriptstyle \eta }$ is constant on alternating vertical infinite strips, that is for all ${\displaystyle \scriptstyle i,j}$:

${\displaystyle \eta (4i,j)=\eta (4i+1,j)=1,\quad \eta (4i+2,j)=\eta (4i+3,j)=0}$

then no transition ever occur, and the process fixates.

(b) Under the assumption of Theorem 3.2, the process does not fixate. To see this, consider the initial configuration ${\displaystyle \scriptstyle \dots 000111\dots }$, in which infinitely many zeros are followed by infinitely many ones. Then only the zero and one at the boundary can flip, so that the configuration will always look the same except that the boundary will move like a simple symmetric random walk. The fact that this random walk is recurrent implies that every site flips infinitely often.

Property 3 indicates that the threshold voter model is quite different from the linear voter model, in that coexistence occurs even in one dimension, provided that the neighbourhood is not too small. The threshold model has a drift toward the "local minority", which is not present in the linear case.

Most proofs of coexistence for threshold voter models are based on comparisons with hybrid model known as the threshold contact process with parameter ${\displaystyle \scriptstyle \lambda >0}$. This is the process on ${\displaystyle \scriptstyle [0,1]^{Z^{d}}}$ with flip rates:

${\displaystyle c(x,\eta )=\left\{{\begin{array}{l}\lambda \quad {\text{if}}\quad \eta (x)=0\quad {\text{and}}|\{y\in x+{\mathcal {N}}:\eta (y)=1\}|\geq T;\\1\quad {\text{if}}\quad \eta (x)=1;\\0\quad {\text{otherwise}}\end{array}}\right.}$

Proposition 3.3

For any ${\displaystyle \scriptstyle d,{\mathcal {N}}}$ and ${\displaystyle \scriptstyle T}$, if the threshold contact process with ${\displaystyle \scriptstyle \lambda =1}$ has a nontrivial invariant measure, then the threshold voter model coexists.

Model with threshold T = 1

The case that ${\displaystyle \scriptstyle T=1}$ is of particular interest because it is the only case in which it is known exactly which models coexist and which models cluster.

In particular, there is interest in a kind of Threshold T=1 model with ${\displaystyle \scriptstyle c(x,\eta )}$ that is given by:

${\displaystyle c(x,\eta )=\left\{{\begin{array}{l}1\quad {\text{if exists one}}\quad y\quad {\text{with}}\quad |x-y|\leq N\quad {\text{and}}\quad \eta (x)\neq \eta (y)\\0\quad {\text{otherwise}}\\\end{array}}\right.}$

${\displaystyle \scriptstyle N}$ can be interpreted as the radius of the neighbourhood ${\displaystyle \scriptstyle {\mathcal {N}}}$; ${\displaystyle \scriptstyle N}$ determines the size of the neighbourhood (i.e., if ${\displaystyle \scriptstyle {\mathcal {N}}_{1}=\{-2,-1,0,1,2\}}$, then ${\displaystyle \scriptstyle N_{1}=2}$; while for ${\displaystyle \scriptstyle {\mathcal {N}}_{2}=\{(0,0),(0,1),(1,0),(0,-1),(-1,0)\}}$, the corresponding ${\displaystyle \scriptstyle N_{2}=1}$).

By Theorem 3.2, the model with ${\displaystyle \scriptstyle d=1}$ and ${\displaystyle \scriptstyle {\mathcal {N}}=\{-1,0,1\}}$ clusters. The following theorem indicates that for all other choices of ${\displaystyle \scriptstyle d}$ and ${\displaystyle \scriptstyle {\mathcal {N}}}$, the model coexists.

Theorem 3.4

Suppose that ${\displaystyle \scriptstyle N\geq 1}$, but ${\displaystyle \scriptstyle (N,d)\neq (1,1)}$. Then the threshold model on ${\displaystyle \scriptstyle Z^{d}}$ with parameter ${\displaystyle \scriptstyle N}$ coexists.

The proof of this theorem is given in a paper named "Coexistence in threshold voter models" by Thomas M. Liggett.