Definition
A voter model is a (continuous time) Markov process
with state space
and transition rates function
, where
is a d-dimensional integer lattice, and
•,•
is assumed to be nonnegative, uniformly bounded and continuous as a function of
in the product topology on
. Each component
is called a configuration. To make it clear that
stands for the value of a site x in configuration
; while
means the value of a site x in configuration
at time
.
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
from 0 to 1 or vice versa is given by a function
of site
. It has the following properties:
for every
if
or if ![{\displaystyle \eta \equiv 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22af8a04129879f85ebd02d74fd88f731f6db7bf)
for every
if
for all ![{\displaystyle y\in Z^{d))](https://wikimedia.org/api/rest_v1/media/math/render/svg/e39ea38e869f065f861719ad8ab16e4e902eb0eb)
if
and ![{\displaystyle \eta (x)=\zeta (x)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3d1e6df0cee25626ce416c4ac16ca2dec6f5bc1)
is invariant under shifts in ![{\displaystyle \scriptstyle Z^{d))](https://wikimedia.org/api/rest_v1/media/math/render/svg/437a060d690693b6c27ff0edfcc530cf4e4ad7cc)
Property (1) says that
and
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),
means
, and
implies
if
, and implies
if
.
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
and
on
or
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
and all initial configurations, then
![{\displaystyle \lim _{t\rightarrow \infty }P[\eta _{t}(x)\neq \eta _{t}(y)]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/acf928873c5896404550174849e22ddfc28afbe5)
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
or
.
The linear voter model
Model description
This section will be dedicated to one of the basic voter models, the Linear Voter Model.
If
•,•
be the transition probabilities for an irreducible random walk on
, then:
![{\displaystyle p(x,y)\geq 0\quad {\text{and))\sum _{y}p(x,y)=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ec7657f2512aa7025a7e47b1d3143c8769a53ac)
Then in Linear voter model, the transition rates are linear functions of
:
![{\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.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c26e6571361107019c197ccdf5a8790454bdf2e)
Or if
indicates that a flip happens at
, then transition rates are simply:
![{\displaystyle \eta \rightarrow \eta _{x}\quad {\text{at rate))\sum _{y:\eta (y)\neq \eta (x)}p(x,y).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8427f57bc27274ba28c4c5041d62555d07501c45)
A process of coalescing random walks
is defined as follows. Here
denotes the set of sites occupied by these random walks at time
. To define
, consider several (continuous time) random walks on
with unit exponential holding times and transition probabilities
•,•
, 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
•,•
.
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),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/46eb385619b98db4ca6a03f0ee20a6ab3d1e8d7e)
where
is the initial configuration of
and
is the initial state of the coalescing random walks
.
Limiting behaviors of linear voter models
Let
be the transition probabilities for an irreducible random walk on
and
, then the duality relation for such linear voter models says that
![{\displaystyle P^{\eta }[\eta _{t}(x)\neq \eta _{t}(y)]=P[\eta (X_{t})\neq \eta (Y_{t})]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b24bafc0ef1743b2886794d1f3b0cdb9165b8dca)
where
and
are (continuous time) random walks on
with
,
, and
is the position taken by the random walk at time
.
and
forms a coalescing random walks described at the end of section 2.1.
is a symmetrized random walk. If
is recurrent and
,
and
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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/74bae0a663137891a0aa2a6a3808fd9d5b9c9bf8)
Therefore, the process clusters.
On the other hand, when
, the system coexists. It is because for
,
is transient, thus there is a positive probability that the random walks never hit, and hence for
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e78af1e931b18286d5ff7753e57a1e6ebdbb1b3)
for some constant
corresponding to the initial distribution.
If
be a symmetrized random walk, then there are the following theorems:
Theorem 2.1
The linear voter model
clusters if
is recurrent, and coexists if
is transient. In particular,
- the process clusters if
and
, or if
and
;
- the process coexists if
.
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
is any translation spatially ergodic and invariant probability measure on the state space
, then
- If
is recurrent, then
;
- If
is transient, then
.
where
is the distribution of
;
means weak convergence,
is a nontrivial extremal invariant measure and
.
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 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))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ad8c2afa07f69a8f6d81445b563df8975a6211e)
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)\}|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/184695a25e0d7c82796c8c87c6b24f7b6fc88c09)
In this case, the process clusters if
, while coexists if
. This dichotomy is closely related to the fact that simple random walk on
is recurrent if
and transient if
.
Clusters in one dimension d = 1
For the special case with
,
and
for each
. From Theorem 2.2,
, thus clustering occurs in this case. The aim of this section is to give a more precise description of this clustering.
As mentioned before, clusters of an
are defined to be the connected components of
or
. The mean cluster size for
is defined to be:
![{\displaystyle C(\eta )=\lim _{n\rightarrow \infty }{\frac {2n}((\text{number of clusters in))[-n,n]))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4426bd90f5c1836e16ed646e675a5c17fdadc484)
provided the limit exists.
Proposition 2.3
Suppose the voter model is with initial distribution
and
is a translation invariant probability measure, then
![{\displaystyle P\left(C(\eta )={\frac {1}{P[\eta _{t}(0)\neq \eta _{t}(1)]))\right)=1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae61446814b3797d9de3369560f580214b634fce)
Occupation time
Define the occupation time functionals of the basic linear voter model as:
![{\displaystyle T_{t}^{x}=\int _{0}^{t}\eta _{s}^{\rho }(x)\mathrm {d} s.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/067662bb3dfc5b8982e0ebe93729b950175738e8)
Theorem 2.4
Assume that for all site x and time t,
, then as
,
almost surely if
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>1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/52ce032e69f83cf8162e0859556467de2ec99405)
The theorem follows when letting
.
The threshold voter model
Model description
This section concentrates on a kind of non-linear voter model, known as the threshold voter model. To define it, let
be a neighbourhood of
that is obtained by intersecting
with any compact, convex, symmetric set in
; in other words,
is assumed to be a finite set that is symmetric with respect to all reflections and irreducible (i.e. the group it generates is
). It can always be assumed that
contains all the unit vectors
. For a positive integer
, the threshold voter model with neighbourhood
and threshold
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.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/048ad97d2a9bbfb88300c7ba75ac14fd835f79db)
Simply put, the transition rate of site
is 1 if the number of sites that do not take the same value is larger or equal to the threshold T. Otherwise, site
stays at the current status and will not flip.
For example, if
,
and
, then the configuration
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,
. 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:
- If
, then the process fixates in the sense that each site flips only finitely often.
- If
and
, then the process clusters.
- If
with
sufficiently small(
) and
sufficiently large, then the process coexists.
Here are two theorems corresponding to properties (1) and (2).
Theorem 3.1
If
, then the process fixates.
Theorem 3.2
The threshold voter model in one dimension (
) with
, clusters.
proof
The idea of the proof is to construct two sequences of random times
,
for
with the following properties:
,
are i.i.d.with
,
are i.i.d.with
,
- the random variables in (b) and (c) are independent of each other,
- event A=
is constant on
, and event A holds for every
.
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 }](https://wikimedia.org/api/rest_v1/media/math/render/svg/2944ef2a1fa5e70663f716f5e34c5aece7d1a700)
Hence,
, so that the process clusters.
Remarks: (a) Threshold models in higher dimensions do not necessarily cluster if
. For example, take
and
. If
is constant on alternating vertical infinite strips, that is for all
:
![{\displaystyle \eta (4i,j)=\eta (4i+1,j)=1,\quad \eta (4i+2,j)=\eta (4i+3,j)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1af257ddb0e7b8069c94be738f3687e33911a11f)
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
, 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
. This is the process on
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.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e8419ee00bf9a08cbb589751bc1bddd76e18044)
Proposition 3.3
For any
and
, if the threshold contact process with
has a nontrivial invariant measure, then the threshold voter model coexists.
Model with threshold T = 1
The case that
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
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.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e25468aeaa013455aaa91a32dba06c552b29c67)
can be interpreted as the radius of the neighbourhood
;
determines the size of the neighbourhood (i.e., if
, then
; while for
, the corresponding
).
By Theorem 3.2, the model with
and
clusters. The following theorem indicates that for all other choices of
and
, the model coexists.
Theorem 3.4
Suppose that
, but
. Then the threshold model on
with parameter
coexists.
The proof of this theorem is given in a paper named "Coexistence in threshold voter models" by Thomas M. Liggett.