Examples: Birth-and-Death Processes; Ising Model

*Birth-and-Death Processes*- The update function
defined in
(88) satisfies the monotonicity condition
(91)
- if the state space can identified with the set equipped with the natural order of the numbers
- and if the simulation matrix
is
*monotonously nondecreasing*with respect to the order , i.e., for arbitrary such that we have

- A whole class of transition matrices
satisfying
the monotonicity condition (100) is given by the
tridiagonal matrices of
*birth-and-death processes*which are of the type

- On the other hand, the update function
defined in (88) is monotonously nonincreasing, see
(98),
- if
is
*monotonously nonincreasing*with respect to , - i.e., if for arbitrary such that we have

- if
is
- It is easy to show that there is no tridiagonal transition matrix
satisfying the condition (101),
i.e., birth-and-death processes are never monotonously
nonincreasing.

- However, condition (101) holds for example for the
following matrix:

- The update function
defined in
(88) satisfies the monotonicity condition
(91)
*Ising Model*- Like for the hard-core model discussed in
Section 3.3.1
- we consider a connected graph with finitely many vertices
- and a certain set of edges , each of them connecting two vertices .

- One of the values and is assigned to each vertex,
- and we consider the state space of all configurations , i.e. for each either or .
- If this is interpreted as an image, is regarded as a white pixel and as a black pixel.

- For each
let the probability
of the
configuration
be given by

for a certain parameter , which is interpreted as ,,inverse temperature'' in physics:

- For (infinite temperature) the distribution given by (102) is the discrete uniform distribution.
- For (low temperature) those configurations possess a large probability that have a small number of connected pairs of vertices being differently colored.
- For (zero temperature) the distribution given by (102) converges to the ,,two point uniform distribution'' ,
- where and denote the (extreme) configurations consisting either only of white or only of black pixels, i.e. either or for all .

- Notice that is an (in general unknown) normalizing
constant where
- The following figure was taken from O. Häggström (2002)
*Finite Markov Chains and Algorithmic Applications*, CU Press, Cambridge.- It illustrates the role of the parameter ,
- i.e., an increase of results in a more pronounced clumping tendency of identically colored pixels.

**Figure 8:**Typical configuration of the Ising model for (upper left corner), (upper right corner), (lower left corner) and (lower right corner)

- Let the simulation matrix
be given
by the Gibbs sampler, i.e., assume that (36) holds,
namely
- where for arbitrary
such that
- By (102) we obtain for
that

where and denote the number of vertices connected to having the values and , respectively.

- where for arbitrary
such that
- For the state space
we define the partial order
- by if for all such that for all ,
- where we assume the elements of the state space
to be indexed in a way ensuring
if
(this is e.g. the case if
is ordered
*lexicographically*).

- Then (103) implies for arbitrary
such that

because for arbitrary such that . - Let the update function
be given by
, where
and for all

- Like for the hard-core model discussed in
Section 3.3.1