<s>
In	O
the	O
mathematical	O
theory	O
of	O
probability	O
,	O
the	O
voter	B-Algorithm
model	I-Algorithm
is	O
an	O
interacting	O
particle	O
system	O
introduced	O
by	O
Richard	O
A	O
.	O
Holley	O
and	O
Thomas	O
M	O
.	O
Liggett	O
in	O
1975	O
.	O
</s>
<s>
Problems	O
involving	O
the	O
voter	B-Algorithm
model	I-Algorithm
will	O
often	O
be	O
recast	O
in	O
terms	O
of	O
the	O
dual	O
system	O
of	O
coalescing	O
Markov	O
chains	O
.	O
</s>
<s>
A	O
voter	B-Algorithm
model	I-Algorithm
is	O
a	O
(	O
continuous	O
time	O
)	O
Markov	O
process	O
with	O
state	O
space	O
and	O
transition	O
rates	O
function	O
,	O
where	O
is	O
a	O
d-dimensional	O
integer	O
lattice	O
,	O
and	O
•	O
,	O
•	O
is	O
assumed	O
to	O
be	O
nonnegative	O
,	O
uniformly	O
bounded	O
and	O
continuous	O
as	O
a	O
function	O
of	O
in	O
the	O
product	O
topology	O
on	O
.	O
</s>
<s>
For	O
voter	B-Algorithm
models	I-Algorithm
,	O
the	O
rate	O
at	O
which	O
there	O
is	O
a	O
flip	O
at	O
from	O
0	O
to	O
1	O
or	O
vice	O
versa	O
is	O
given	O
by	O
a	O
function	O
of	O
site	O
.	O
</s>
<s>
Therefore	O
,	O
a	O
voter	B-Algorithm
model	I-Algorithm
has	O
two	O
trivial	O
extremal	O
stationary	O
distributions	O
,	O
the	O
point-masses	O
and	O
on	O
or	O
respectively	O
,	O
which	O
represent	O
consensus	O
.	O
</s>
<s>
This	O
section	O
will	O
be	O
dedicated	O
to	O
one	O
of	O
the	O
basic	O
voter	B-Algorithm
models	I-Algorithm
,	O
the	O
Linear	O
Voter	B-Algorithm
Model	I-Algorithm
.	O
</s>
<s>
Then	O
in	O
Linear	O
voter	B-Algorithm
model	I-Algorithm
,	O
the	O
transition	O
rates	O
are	O
linear	O
functions	O
of	O
:	O
</s>
<s>
The	O
concept	O
of	O
Duality	O
is	O
essential	O
for	O
analysing	O
the	O
behavior	O
of	O
the	O
voter	B-Algorithm
models	I-Algorithm
.	O
</s>
<s>
The	O
linear	O
voter	B-Algorithm
models	I-Algorithm
satisfy	O
a	O
very	O
useful	O
form	O
of	O
duality	O
,	O
known	O
as	O
coalescing	O
duality	O
,	O
which	O
is	O
:	O
</s>
<s>
The	O
linear	O
voter	B-Algorithm
model	I-Algorithm
clusters	O
if	O
is	O
recurrent	O
,	O
and	O
coexists	O
if	O
is	O
transient	O
.	O
</s>
<s>
Remarks	O
:	O
To	O
contrast	O
this	O
with	O
the	O
behavior	O
of	O
the	O
threshold	O
voter	B-Algorithm
models	I-Algorithm
that	O
will	O
be	O
discussed	O
in	O
next	O
section	O
,	O
note	O
that	O
whether	O
the	O
linear	O
voter	B-Algorithm
model	I-Algorithm
clusters	O
or	O
coexists	O
depends	O
almost	O
exclusively	O
on	O
the	O
dimension	O
of	O
the	O
set	O
of	O
sites	O
,	O
rather	O
than	O
on	O
the	O
size	O
of	O
the	O
range	O
of	O
interaction	O
.	O
</s>
<s>
One	O
of	O
the	O
interesting	O
special	O
cases	O
of	O
the	O
linear	O
voter	B-Algorithm
model	I-Algorithm
,	O
known	O
as	O
the	O
basic	O
linear	O
voter	B-Algorithm
model	I-Algorithm
,	O
is	O
that	O
for	O
state	O
space	O
:	O
</s>
<s>
Define	O
the	O
occupation	O
time	O
functionals	O
of	O
the	O
basic	O
linear	O
voter	B-Algorithm
model	I-Algorithm
as	O
:	O
</s>
<s>
This	O
section	O
,	O
concentrates	O
on	O
a	O
kind	O
of	O
non-linear	O
voter	B-Algorithm
models	I-Algorithm
,	O
known	O
as	O
the	O
threshold	O
voter	B-Algorithm
model	I-Algorithm
.	O
</s>
<s>
For	O
a	O
positive	O
integer	O
,	O
the	O
threshold	O
voter	B-Algorithm
model	I-Algorithm
with	O
neighbourhood	O
and	O
threshold	O
is	O
the	O
one	O
with	O
rate	O
function	O
:	O
</s>
<s>
If	O
a	O
threshold	O
voter	B-Algorithm
model	I-Algorithm
does	O
not	O
fixate	O
,	O
the	O
process	O
should	O
be	O
expected	O
to	O
will	O
coexist	O
for	O
small	O
threshold	O
and	O
cluster	O
for	O
large	O
threshold	O
,	O
where	O
large	O
and	O
small	O
are	O
interpreted	O
as	O
being	O
relative	O
to	O
the	O
size	O
of	O
the	O
neighbourhood	O
,	O
.	O
</s>
<s>
The	O
threshold	O
voter	B-Algorithm
model	I-Algorithm
in	O
one	O
dimension	O
(	O
)	O
with	O
,	O
clusters	O
.	O
</s>
<s>
Property	O
3	O
indicates	O
that	O
the	O
threshold	O
voter	B-Algorithm
model	I-Algorithm
is	O
quite	O
different	O
from	O
the	O
linear	O
voter	B-Algorithm
model	I-Algorithm
,	O
in	O
that	O
coexistence	O
occurs	O
even	O
in	O
one	O
dimension	O
,	O
provided	O
that	O
the	O
neighbourhood	O
is	O
not	O
too	O
small	O
.	O
</s>
<s>
Most	O
proofs	O
of	O
coexistence	O
for	O
threshold	O
voter	B-Algorithm
models	I-Algorithm
are	O
based	O
on	O
comparisons	O
with	O
hybrid	O
model	O
known	O
as	O
the	O
threshold	O
contact	B-Algorithm
process	I-Algorithm
with	O
parameter	O
.	O
</s>
<s>
For	O
any	O
and	O
,	O
if	O
the	O
threshold	O
contact	B-Algorithm
process	I-Algorithm
with	O
has	O
a	O
nontrivial	O
invariant	O
measure	O
,	O
then	O
the	O
threshold	O
voter	B-Algorithm
model	I-Algorithm
coexists	O
.	O
</s>
<s>
The	O
proof	O
of	O
this	O
theorem	O
is	O
given	O
in	O
a	O
paper	O
named	O
"	O
Coexistence	O
in	O
threshold	O
voter	B-Algorithm
models	I-Algorithm
"	O
by	O
Thomas	O
M	O
.	O
Liggett	O
.	O
</s>
