投票者模型

In the mathematical theory of probability, the voter model is an interacting particle system introduced by Richard A. Holley and Thomas M. Liggett in 1975.[1]

In the mathematical theory of probability, the voter model is an interacting particle system introduced by Richard A. Holley and Thomas M. Liggett in 1975.

voter model coexists on the graph with two clusters

thumb|right|voter model coexists on the graph with two clusters

One can imagine that there is a "voter" at each point on a connected graph, where the connections indicate that there is some form of interaction between a pair of voters (nodes). The opinions of any given voter on some issue changes at random times under the influence of opinions of his neighbours. A voter's opinion at any given time can take one of two values, labelled 0 and 1. At random times, a random individual is selected and that voter's opinion is changed according to a stochastic rule. Specifically, for one of the chosen voter's neighbors is chosen模板:Clarify according to a given set of probabilities and that individual's opinion is transferred to the chosen voter.

One can imagine that there is a "voter" at each point on a connected graph, where the connections indicate that there is some form of interaction between a pair of voters (nodes). The opinions of any given voter on some issue changes at random times under the influence of opinions of his neighbours. A voter's opinion at any given time can take one of two values, labelled 0 and 1. At random times, a random individual is selected and that voter's opinion is changed according to a stochastic rule. Specifically, for one of the chosen voter's neighbors is chosen according to a given set of probabilities and that individual's opinion is transferred to the chosen voter.

An alternative interpretation is in terms of spatial conflict. Suppose two nations control the areas (sets of nodes) labelled 0 or 1. A flip from 0 to 1 at a given location indicates an invasion of that site by the other nation.

An alternative interpretation is in terms of spatial conflict. Suppose two nations control the areas (sets of nodes) labelled 0 or 1. A flip from 0 to 1 at a given location indicates an invasion of that site by the other nation.

Note that only one flip happens each time. Problems involving the voter model will often be recast in terms of the dual system模板:Clarify of coalescing模板:Clarify Markov chains. Frequently, these problems will then be reduced to others involving independent Markov chains.

Note that only one flip happens each time. Problems involving the voter model will often be recast in terms of the dual system of coalescing Markov chains. Frequently, these problems will then be reduced to others involving independent Markov chains.

= 定义 =

A voter model is a (continuous time) Markov process $\displaystyle{ \eta_t }$ with state space $\displaystyle{ S=\{0,1\}^{Z^d} }$ and transition rates function $\displaystyle{ c(x,\eta) }$, where $\displaystyle{ Z^d }$ is a d-dimensional integer lattice, and $\displaystyle{ c( }$•,•$\displaystyle{ ) }$ is assumed to be nonnegative, uniformly bounded and continuous as a function of $\displaystyle{ \eta }$ in the product topology on $\displaystyle{ S }$. Each component $\displaystyle{ \eta \in S }$ is called a configuration. To make it clear that $\displaystyle{ \eta(x) }$ stands for the value of a site x in configuration $\displaystyle{ \eta(.) }$; while $\displaystyle{ \eta_t(x) }$ means the value of a site x in configuration $\displaystyle{ \eta(.) }$ at time $\displaystyle{ t }$.

A voter model is a (continuous time) Markov process \eta_t with state space S=\{0,1\}^{Z^d} and transition rates function c(x,\eta) , where Z^d is a d-dimensional integer lattice, and c( •,• ) is assumed to be nonnegative, uniformly bounded and continuous as a function of \eta in the product topology on S . Each component \eta \in S is called a configuration. To make it clear that \eta(x) stands for the value of a site x in configuration \eta(.) ; while \eta_t(x) means the value of a site x in configuration \eta(.) at time t.

The dynamic of the process are specified by the collection of transition rates. For voter models, the rate at which there is a flip at $\displaystyle{ \scriptstyle x }$ from 0 to 1 or vice versa is given by a function $\displaystyle{ c(x,\eta) }$ of site $\displaystyle{ x }$. It has the following properties:

The dynamic of the process are specified by the collection of transition rates. For voter models, the rate at which there is a flip at \scriptstyle x from 0 to 1 or vice versa is given by a function c(x,\eta) of site x . It has the following properties:

1. $\displaystyle{ c(x,\eta)=0 }$ for every $\displaystyle{ x \in Z^d }$ if $\displaystyle{ \eta \equiv 0 }$ or if $\displaystyle{ \eta \equiv 1 }$
2. $\displaystyle{ c(x,\eta)=c(x,\zeta) }$ for every $\displaystyle{ x \in Z^d }$ if $\displaystyle{ \eta(y)+\zeta(y)=1 }$ for all $\displaystyle{ y \in Z^d }$
3. $\displaystyle{ c(x,\eta)\leq c(x,\zeta) }$ if $\displaystyle{ \eta\leq \zeta }$ and $\displaystyle{ \eta(x)=\zeta(x)=0 }$
4. $\displaystyle{ c(x,\eta) }$ is invariant under shifts in $\displaystyle{ \scriptstyle Z^d }$
1. c(x,\eta)=0 for every x \in Z^d if \eta \equiv 0 or if \eta \equiv 1
2. c(x,\eta)=c(x,\zeta) for every x \in Z^d if \eta(y)+\zeta(y)=1 for all y \in Z^d
3. c(x,\eta)\leq c(x,\zeta) if \eta\leq \zeta and \eta(x)=\zeta(x)=0
4. c(x,\eta) is invariant under shifts in \scriptstyle Z^d
1. c (x，eta) = 0对于 z ^ d 中的每个 x，如果 eta equiv0或者如果 eta equiv1 # c (x，eta) = c (x，zeta)对于 z ^ d 中的每个 x，如果 eta (y) + zeta (y) = 1对于 z ^ d c (x，eta)中的所有 y，如果 eta lezeta 和 eta (x) = zeta (x) = 0 # c (x，eta 不变式 d 处于移位之下

Property (1) says that $\displaystyle{ \eta\equiv 0 }$ and $\displaystyle{ \eta\equiv 1 }$ are fixed points for the evolution. (2) indicates that the evolution is unchanged by interchanging the roles of 0's and 1's. In property (3), $\displaystyle{ \eta\leq \zeta }$ means $\displaystyle{ \forall x,\eta(x)\leq\zeta(x) }$, and $\displaystyle{ \eta \leq \zeta }$ implies $\displaystyle{ c(x,\eta)\leq c(x,\zeta) }$ if $\displaystyle{ \eta(x)=\zeta(x)=0 }$, and implies $\displaystyle{ c(x,\eta)\geq c(x,\zeta) }$ if $\displaystyle{ \eta(x)=\zeta(x)=1 }$.

Property (1) says that \eta\equiv 0 and \eta\equiv 1 are fixed points for the evolution. (2) indicates that the evolution is unchanged by interchanging the roles of 0's and 1's. In property (3), \eta\leq \zeta means \forall x,\eta(x)\leq\zeta(x) , and \eta \leq \zeta implies c(x,\eta)\leq c(x,\zeta) if \eta(x)=\zeta(x)=0 , and implies c(x,\eta)\geq c(x,\zeta) if \eta(x)=\zeta(x)=1 .

Clustering and coexistence

The interest in is the limiting behavior of the models. Since the flip rates of a site depends 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.

The interest in is the limiting behavior of the models. Since the flip rates of a site depends 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 \scriptstyle \delta_0 and \scriptstyle \delta_1 on \scriptstyle \eta \equiv 0 or \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 \scriptstyle x,y\in Z^d and all initial configurations, then

\lim_{t\rightarrow \infty}P[\eta_t(x)\neq\eta_t(y)]=0

It is said that clustering occurs.

= = = 群集和共存 = = = = 关注的是模型的限制行为。由于一个网站的翻转率取决于它的邻居，很明显，当所有网站的价值相同时，整个系统将永远停止变化。因此，选民模型有两个平凡的极值平稳分布，点质量 scriptstyle delta _ 0和 scriptstyle delta _ 1分别在 scriptstyle eta equv 0和 scriptstyle eta equv 1上，代表共识。要讨论的主要问题是，是否还有其他的问题，这些问题将表现出不同意见在平衡中的共存。我们说，如果有一个静态分布，它集中在具有无穷多个0和1的构型上，那么共存就发生了。另一方面，如果对于所有的 scriptstyle x，y 在 z ^ d 和所有的初始配置，那么: lim { t right tarrow infty } p [ eta _ t (x) neq eta _ t (y)] = 0据说发生了聚类。

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 $\displaystyle{ \scriptstyle \{x:\eta(x)=1\} }$.

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

= 线性选民模型 =

Model description

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

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

= = = 模型描述 = = 本节将致力于一个基本的选民模型，线性选民模型。

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 $\displaystyle{ \scriptstyle \eta }$:

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

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

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

 c(x,\eta)= \left\{
\begin{array}{l}
\end{array} \right.


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

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

Or if \scriptstyle \eta_x indicates that a flip happens at \scriptstyle x, then transition rates are simply:

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 $\displaystyle{ \scriptstyle p( }$•,•$\displaystyle{ \scriptstyle ) }$ .

A process of coalescing random walks \scriptstyle A_t\subset Z^d is defined as follows. Here \scriptstyle A_t denotes the set of sites occupied by these random walks at time \scriptstyle t . To define \scriptstyle A_t , consider several (continuous time) random walks on \scriptstyle Z^d with unit exponential holding times and transition probabilities \scriptstyle p( •,•\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 \scriptstyle p( •,•\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 $\displaystyle{ \scriptstyle A_t }$.

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:

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

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

P^{\eta}[\eta_t(x)\neq\eta_t(y)]=P[\eta(X_t)\neq\eta(Y_t)]

where \scriptstyle X_t and \scriptstyle Y_t are (continuous time) random walks on \scriptstyle Z^d with \scriptstyle X_0=x , \scriptstyle Y_0=y , and \scriptstyle \eta(X_t) is the position taken by the random walk at time \scriptstyle t . \scriptstyle X_t and \scriptstyle Y_t forms a coalescing random walks described at the end of section 2.1. \scriptstyle X(t)-Y(t) is a symmetrized random walk. If \scriptstyle X(t)-Y(t) is recurrent and \scriptstyle d\leq 2 , \scriptstyle X_t and \scriptstyle Y_t will hit eventually with probability 1, and hence

Therefore, the process clusters.

= = = = = = = 设 scriptstyle p (x，y)为 scriptstyle z ^ d 和 scriptstyle p (x，y) = p (0，x-y)上不可约随机游动的转移概率,然后这种线性选民模型的对偶关系说，scriptstyle forall eta in s = {0,1} ^ { z ^ d } : p ^ { eta }[ eta _ t (x) neq eta _ t (y)] = p [ eta (x _ t) neq _ eta (y _ t)]其中 scriptstyle x _ t 和 scriptstyle y _ t 是(连续时间) scriptstyle z ^ d 上的随机游动，并且 scriptstyle x _ 0 = x,scriptstyle y _ 0 = y，scriptstyle eta (x _ t)是当时随机漫步 scriptstyle t 所采取的位置。脚本风格 x _ t 和脚本风格 y _ t 在第2.1节末尾描述了随机游动的结合。Scriptstyle x (t)-y (t)是对称化的随机漫步。如果 scriptstyle x (t)-y (t)是反复出现的，scriptstyle d leq 2，scriptstyle x _ t 和 scriptstyle y _ t 最终将以1的概率出现，因此: p ^ { eta }[ eta _ t (x) neq eta _ t (y)] = p [ eta (x _ t) neq eta (y _ t)] leq p [ x _ t neq _ t ] right tarrow 0 quad text { as } t to 0。

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 $\displaystyle{ \scriptstyle x\neq y }$

$\displaystyle{ \lim_{t\rightarrow\infty}P[\eta_t(x)\neq\eta_t(y)]=C\lim_{t\rightarrow\infty}P[X_t\neq Y_t]\gt 0 }$

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

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

\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 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:

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

Theorem 2.1

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,

The linear voter model \scriptstyle \eta_t clusters if \scriptstyle \tilde{X}_t is recurrent, and coexists if \scriptstyle \tilde{X}_t is transient. In particular,

1. the process clusters if $\displaystyle{ \scriptstyle d=1 }$ and $\displaystyle{ \scriptstyle \sum_x |x|p(0,x)\le \infty }$, or if $\displaystyle{ \scriptstyle d=2 }$ and $\displaystyle{ \scriptstyle \sum_x |x|^2p(0,x)\le\infty }$;
2. the process coexists if $\displaystyle{ \scriptstyle d \geq 3 }$.
1. the process clusters if \scriptstyle d=1 and \scriptstyle \sum_x |x|p(0,x)\le \infty , or if \scriptstyle d=2 and \scriptstyle \sum_x |x|^2p(0,x)\le\infty ;
2. the process coexists if \scriptstyle d \geq 3 .
1. 如果 scriptstyle d = 1和 scriptstyle sum _ x | x | p (0，x) le infty，或者如果 scriptstyle d = 2和 scriptstyle sum _ x | x | ^ 2p (0，x) le infty，进程集群; # 如果 scriptstyle d geq 3，进程共存。

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.

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

Theorem 2.2 Suppose \scriptstyle \mu is any translation spatially ergodic and invariant probability measure on the state space \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 $\displaystyle{ \scriptstyle \mu S(t)\Rightarrow \mu_\rho }$.
1. If \scriptstyle \tilde{X}_t is recurrent, then \scriptstyle \mu S(t)\Rightarrow \rho\delta_1+(1-\rho)\delta_0\quad\text{as}\quad t\to\infty ;
2. If \scriptstyle \tilde{X}_t is transient, then \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 $\displaystyle{ \scriptstyle \rho=\mu(\{\eta:\eta(x)=1\}) }$.

where \scriptstyle \mu S(t) is the distribution of \scriptstyle \eta_t ; \scriptstyle \Rightarrow means weak convergence, \scriptstyle \mu_{\rho} is a nontrivial extremal invariant measure and \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 $\displaystyle{ \scriptstyle \{0,1\}^{Z^d} }$:

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

= = = = 一个特殊的线性投票人模型 = = = = 线性投票人模型的一个有趣的特殊情况，被称为基本线性投票人模型，是国家空间的脚本风格{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 $\displaystyle{ \scriptstyle d\leq2 }$ and transient if $\displaystyle{ \scriptstyle d\geq 3 }$.

 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

\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 \scriptstyle d\leq 2 , while coexists if \scriptstyle d\geq 3 . This dichotomy is closely related to the fact that simple random walk on \scriptstyle Z^d is recurrent if \scriptstyle d\leq2 and transient if \scriptstyle d\geq 3 .

P (x，y) = begin { cases }1/2d & text { if } | 我们会找到他的= 1 text { and } eta (x) neq eta (y)[8 pt ]0 & text { otherwise } end { cases }所以: 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)} | 在这种情况下，进程集群如果 scriptstyle d leq 2，而共存如果 scriptstyle d geq 3。这种二分法与这样一个事实密切相关: 如果 scriptstyle d leq2，那么在 scriptstyle z ^ d 上简单的随机漫步是反复出现的; 如果 scriptstyle d geq 3，那么简单的随机漫步是暂时的。

Clusters in one dimension d = 1

For the special case with $\displaystyle{ \scriptstyle d=1 }$, $\displaystyle{ \scriptstyle S=Z^1 }$ and $\displaystyle{ \scriptstyle p(x,x+1)=p(x,x-1)=\frac{1}{2} }$ for each $\displaystyle{ \scriptstyle x }$. From Theorem 2.2, $\displaystyle{ \scriptstyle \mu S(t)\Rightarrow \rho\delta_1+(1-\rho)\delta_0 }$, thus clustering occurs in this case. The aim of this section is to give a more precise description of this clustering.

For the special case with \scriptstyle d=1 , \scriptstyle S=Z^1 and \scriptstyle p(x,x+1)=p(x,x-1)=\frac{1}{2} for each \scriptstyle x . From Theorem 2.2, \scriptstyle \mu S(t)\Rightarrow \rho\delta_1+(1-\rho)\delta_0 , thus clustering occurs in this case. The aim of this section is to give a more precise description of this clustering.

= = = = = 一维集群 d = 1 = = = = = = 对于特殊情况，对于每个 scriptstyle x，scriptstyle s = z ^ 1和 scriptstyle p (x，x + 1) = p (x，x-1) = frac {1}{2}。根据定理2.2，scriptstyle mu s (t) right tarrow rho delta _ 1 + (1-rho) delta _ 0，因此在这种情况下发生了聚类。本节的目的是更精确地描述这种集群。

As mentioned before, clusters of an $\displaystyle{ \scriptstyle \eta }$ are defined to be the connected components of $\displaystyle{ \scriptstyle \{x:\eta(x)=0\} }$ or $\displaystyle{ \scriptstyle \{x:\eta(x)=1\} }$. The mean cluster size for $\displaystyle{ \scriptstyle \eta }$ is defined to be:

$\displaystyle{ C(\eta)=\lim_{n\rightarrow\infty}\frac{2n}{\text{number of clusters in} [-n,n]} }$

provided the limit exists.

As mentioned before, clusters of an \scriptstyle \eta are defined to be the connected components of \scriptstyle \{x:\eta(x)=0\} or \scriptstyle \{x:\eta(x)=1\} . The mean cluster size for \scriptstyle \eta is defined to be:

C(\eta)=\lim_{n\rightarrow\infty}\frac{2n}{\text{number of clusters in} [-n,n]}

provided the limit exists.

Proposition 2.3

Proposition 2.3

Suppose the voter model is with initial distribution $\displaystyle{ \scriptstyle \mu }$ and $\displaystyle{ \scriptstyle \mu }$ is a translation invariant probability measure, then

$\displaystyle{ P\left(C(\eta)=\frac{1}{P[\eta_t(0)\neq \eta_t(1)]}\right)=1. }$

Suppose the voter model is with initial distribution \scriptstyle \mu and \scriptstyle \mu is a translation invariant probability measure, then

P\left(C(\eta)=\frac{1}{P[\eta_t(0)\neq \eta_t(1)]}\right)=1.

Occupation time

Define the occupation time functionals of the basic linear voter model as:

$\displaystyle{ T_t^x=\int_0^t \eta^\rho_s(x)\mathrm{d}s. }$

Define the occupation time functionals of the basic linear voter model as:

T_t^x=\int_0^t \eta^\rho_s(x)\mathrm{d}s.

= = = = 占用时间 = = = 定义基本线性选民模型的占用时间泛函为: t _ t ^ x = int _ 0 ^ t eta ^ rho _ s (x) mathrm { d }。

Theorem 2.4

Theorem 2.4

Assume that for all site x and time t, $\displaystyle{ \scriptstyle P(\eta_t(x)=1)=\rho }$, then as $\displaystyle{ \scriptstyle t\rightarrow \infty }$, $\displaystyle{ \scriptstyle T_t^x/t\rightarrow \rho }$ almost surely if $\displaystyle{ \scriptstyle d\geq 2 }$

Assume that for all site x and time t, \scriptstyle P(\eta_t(x)=1)=\rho, then as \scriptstyle t\rightarrow \infty , \scriptstyle T_t^x/t\rightarrow \rho almost surely if \scriptstyle d\geq 2

proof

proof

By Chebyshev's inequality and the Borel–Cantelli lemma, there is the equation below:

$\displaystyle{ P\left(\frac{\rho}{r}\leq \lim \inf_{t\rightarrow\infty}\frac{T_t}{t}\leq\lim\sup_{t\rightarrow\infty}\frac{T_t}{t}\leq \rho r\right)=1; \quad\forall r\gt 1 }$

The theorem follows when letting $\displaystyle{ \scriptstyle r\searrow 1 }$.

By Chebyshev's inequality and the Borel–Cantelli lemma, there is the equation below:

P\left(\frac{\rho}{r}\leq \lim \inf_{t\rightarrow\infty}\frac{T_t}{t}\leq\lim\sup_{t\rightarrow\infty}\frac{T_t}{t}\leq \rho r\right)=1; \quad\forall r>1

The theorem follows when letting \scriptstyle r\searrow 1 .

= 阈值选民模型 =

Model description

This section, concentrates on a kind of non-linear voter models, 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 word, $\displaystyle{ \scriptstyle \mathcal{N} }$ 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:

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

= = = 模型描述 = = 本节集中讨论一种非线性选民模型，即阈值选民模型。为了定义它，让 scriptstyle 数学{ n }成为 z ^ d 中 scriptstyle 0的邻域，这个邻域是通过将 scriptstyle z ^ d 与 scriptstyle r ^ d 中的任意紧凑、凸、对称集合相交而得到的; 换句话说，scriptstyle 数学{ n }被假定为对所有反射和不可约的对称的有限集合。它生成的组是 scriptstyle z ^ d)。可以假定 scriptstyle 数学{ n }包含所有单元向量 scriptstyle (1,0,0，点，0)、 dots (0，点，0,1)。对于正整数 scriptstyle t，带有邻里 scriptstyle 数学{ n }和阈值 scriptstyle t 的阈值选民模型是具有 rate 函数的模型:

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


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.

Simply put, the transition rate of site \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 \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 $\displaystyle{ \scriptstyle \dots1\quad 1\quad 0\quad 0\quad 1\quad 1\quad 0\quad 0\dots }$ 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 $\displaystyle{ \scriptstyle T\gt \frac{|\mathcal{N}|-1}{2} }$, 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($\displaystyle{ \scriptstyle \theta\lt \frac{1}{4} }$) and $\displaystyle{ \scriptstyle |\mathcal{N}| }$ sufficiently large, then the process coexists.

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

= = = 限制阈值选民模型的行为 = = = 如果阈值选民模型不固定，那么对于小阈值进程应该共存，对于大阈值进程应该聚类，其中大和小被解释为相对于邻居的大小，scriptstyle | math{ n } | 。我们的直觉是，有一个小的阈值使得翻转很容易发生，所以很可能在任何时候都会有很多0和1。以下是三个主要结果: # 如果 scriptstyle t > frac { | mathcal { n } |-1}{2} ，那么这个进程固定在每个站点只有限次翻转的意义上。# 如果 scriptstyle d = 1和 scriptstyle t = frac { | mathcal { n } |-1}{2} ，那么进程集群。如果 scriptstyle t = theta | mathcal { n } | with scriptstyle theta 充分小(scriptstyle theta < frac {1}{4})和 scriptstyle | mathcal { n } | 足够大，那么进程共存。

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

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

Theorem 3.1

Theorem 3.1

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

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

Theorem 3.2

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.

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

proof

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:

The idea of the proof is to construct two sequences of random times \scriptstyle U_n , \scriptstyle V_n for \scriptstyle n\geq 1 with the following properties:

1. $\displaystyle{ \scriptstyle 0=V_0\lt U_1\lt V_1\lt U_2\lt V_2\lt \dots }$,
2. $\displaystyle{ \scriptstyle \{U_{k+1}-V_k,k\geq0\} }$ are i.i.d.with $\displaystyle{ \scriptstyle \mathrm{E}(U_{k+1}-V_k)\lt \infty }$,
3. $\displaystyle{ \scriptstyle \{V_{k}-U_k,k\geq1\} }$ 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 $\displaystyle{ \scriptstyle t \in \cup_{k=1}^\infty [U_k,V_k] }$.
1. \scriptstyle 0=V_0<U_1<V_1<U_2<V_2<\dots ,
2. \scriptstyle \{U_{k+1}-V_k,k\geq0\} are i.i.d.with \scriptstyle \mathrm{E}(U_{k+1}-V_k)<\infty ,
3. \scriptstyle \{V_{k}-U_k,k\geq1\} are i.i.d.with \scriptstyle \mathrm{E}(V_{k}-U_k)=\infty ,
4. the random variables in (b) and (c) are independent of each other,
5. event A=\scriptstyle \{\eta_t(.) is constant on \scriptstyle \{-T,\dots,T\}\} , and event A holds for every \scriptstyle t \in \cup_{k=1}^\infty [U_k,V_k] .
1. \scriptstyle 0=V_0<U_1<V_1<U_2<V_2<\dots ,
2. \scriptstyle \{U_{k+1}-V_k,k\geq0\} are i.i.d.with \scriptstyle \mathrm{E}(U_{k+1}-V_k)<\infty ,
3. \scriptstyle \{V_{k}-U_k,k\geq1\} are i.i.d.with \scriptstyle \mathrm{E}(V_{k}-U_k)=\infty ,
4. the random variables in (b) and (c) are independent of each other,
5. event A=\scriptstyle \{\eta_t(.)对于 cup _ { k = 1} ^ infty [ u _ k，v _ k ]中的每个 scriptstyle t，事件 a 都是常量。

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.

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

Hence,\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.

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

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 $\displaystyle{ \scriptstyle \dots 0 0 0 1 1 1 \dots }$, 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.

(b) Under the assumption of Theorem 3.2, the process does not fixate. To see this, consider the initial configuration \scriptstyle \dots 0 0 0 1 1 1 \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.

(b)在定理3.2的假设下，过程不固定。要看到这一点，考虑初始配置 scriptstyle dots 0011 dots，其中无限多个零后面跟着无限多个1。然后，只有边界处的0和1可以翻转，这样配置看起来总是一样的，除了边界会像一个简单的对称随机游走一样移动。事实上，这种随机漫步是反复出现的，这意味着每个站点都会无限频繁地翻转。

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.

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 $\displaystyle{ \scriptstyle \lambda\gt 0 }$. This is the process on $\displaystyle{ \scriptstyle [0,1]^{Z^d} }$ with flip rates:

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

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

c(x,\eta)= \left\{
\begin{array}{l}


\end{array}\right.

Proposition 3.3

Proposition 3.3

3.3号提案

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.

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

Model with threshold T = 1

= = = 具有阈值 t = 1 = = = 的模型

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.

The case that \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.

Scriptstyle t = 1的情况特别有趣，因为它是唯一知道哪些模型共存和哪些模型集群的情况。

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

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

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

 c(x,\eta)= \left\{
\begin{array}{l}
\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 $\displaystyle{ \scriptstyle N_2=1 }$).

\scriptstyle N can be interpreted as the radius of the neighbourhood \scriptstyle \mathcal{N} ; \scriptstyle N determines the size of the neighbourhood (i.e., if \scriptstyle \mathcal{N}_1=\{-2,-1,0,1,2\} , then \scriptstyle N_1=2 ; while for \scriptstyle \mathcal{N}_2=\{(0,0),(0,1),(1,0),(0,-1),(-1,0)\} , the corresponding \scriptstyle N_2=1 ).

Scriptstyle n 可以解释为邻域 scriptstyle 数学{ n }的半径; scriptstyle n 确定邻域的大小(即，如果 scriptstyle 数学{ n }1 = {-2,-1,0,1,2} ，那么 scriptstyle n _ 1 = 2; 而对于 scriptstyle 数学{ n } _ 2 = {(0,0) ，(0,1) ，(1,0) ，(0,-1) ，(- 1,0)} ，相应的 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.

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

Theorem 3.4

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.

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

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

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

= 参见 =

• Probabilistic Cellular Automata
• sequential dynamical system
• contact process

• 概率细胞自动机
• 顺序动力系统
• 接触过程

= 笔记 =

1. Holley, Richard A.; Liggett, Thomas M. (1975). "Ergodic Theorems for Weakly Interacting Infinite Systems and the Voter Model". The Annals of Probability (in English). 3 (4): 643–663. doi:10.1214/aop/1176996306. ISSN 0091-1798.

= 参考文献 =

• Thomas M. Liggett, "Stochastic Interacting Systems: Contact, Voter and Exclusion Processes", Springer-Verlag, 1999.

• 托马斯 · m · 利格特,”随机相互作用系统: 接触、选民和排除过程”，Springer-Verlag，1999。

Category:Stochastic models Category:Lattice models

This page was moved from wikipedia:en:Voter model. Its edit history can be viewed at 投票者模型/edithistory