Construction of an irreducible Markov chain in the Ising model. Construction of an irreducible Markov Chain is a mathematical method used to prove results related the changing of magnetic materials in the Ising model, enabling the study of phase transitions and critical phenomena. 

The Ising model, a mathematical model in statistical mechanics, is utilized to study magnetic phase transitions and is a fundamental model of interacting systems. Constructing an irreducible Markov chain within a finite Ising model is essential for overcoming computational challenges encountered when achieving exact goodness-of-fit tests with Markov chain Monte Carlo (MCMC) methods.

Markov bases
In the context of the Ising model, a Markov basis is a set of integer vectors that enables the construction of an irreducible Markov chain. Every integer vector  can be uniquely decomposed as , where  and  are non-negative vectors. A Markov basis  satisfies the following conditions:

(i) For all , there must be  and .

(ii) For any  and any , there always exist  satisfy:

 

and

 

for l = 1,...,k. 

The element of  is moved. An aperiodic, reversible, and irreducible Markov Chain can then be obtained using Metropolis–Hastings algorithm.

Persi Diaconis and Bernd Sturmfels showed that (1) a Markov basis can be defined algebraically as an Ising model  and (2) any generating set for the ideal , is a Markov basis for the Ising model.

Construction of an irreducible Markov Chain
To obtain uniform samples from  and avoid inaccurate p-values, it is necessary to construct an irreducible Markov chain without modifying the algorithm proposed by Diaconis and Sturmfels.

A simple swap  of the form , where  is the canonical basis vector, changes the states of two lattice points in y. The set Z denotes the collection of simple swaps. Two configurations  are -connected by Z if there exists a path between y and y′ consisting of simple swaps .

The algorithm proceeds as follows:
 

with

 

for 

The algorithm can now be described as:

(i) Start with the Markov chain in a configuration 

(ii) Select  at random and let .

(iii) Accept  if ; otherwise remain in y.

Although the resulting Markov Chain possibly cannot leave the initial state, the problem does not arise for a 1-dimensional Ising model. In higher dimensions, this problem can be overcome by using the Metropolis-Hastings algorithm in the smallest expanded sample space .

Irreducibility in the 1-Dimensional Ising Model
The proof of irreducibility in the 1-dimensional Ising model requires two lemmas.

Lemma 1: The max-singleton configuration of  for the 1-dimension Ising model is unique (up to location of its connected components) and consists of  singletons and one connected component of size .

Lemma 2: For  and , let  denote the unique max-singleton configuration. There exists a sequence  such that:

 

and

 

for 

Since  is the smallest expanded sample space which contains , any two configurations in  can be connected by simple swaps Z without leaving . This is proved by Lemma 2, so one can achieve the irreducibility of a Markov chain based on simple swaps for the 1-dimension Ising model.

It is also possible to get the same conclusion for a dimension 2 or higher Ising model using the same steps outlined above.

References

Lattice models
Markov chain Monte Carlo