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.[1] 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

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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[2] and (2) any generating set for the ideal  , is a Markov basis for the Ising model.[3]

Construction of an irreducible Markov Chain

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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  .[4]

Irreducibility in the 1-Dimensional Ising Model

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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.[5]

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

References

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  1. ^ Kannan, Ravi; Mahoney, Michael W.; Montenegro, Ravi (2003). "Rapid mixing of several Markov chains for a hard-core model". In Ibaraki, Toshihide; Katoh, Naoki; Ono, Hirotaka (eds.). Algorithms and Computation, 14th International Symposium, ISAAC 2003, Kyoto, Japan, December 15-17, 2003, Proceedings. Lecture Notes in Computer Science. Vol. 2906. Springer. pp. 663–675. doi:10.1007/978-3-540-24587-2_68.
  2. ^ Diaconis, Persi; Sturmfels, Bernd (February 1998). "Algebraic algorithms for sampling from conditional distributions". The Annals of Statistics. 26 (1): 363–397. CiteSeerX 10.1.1.28.6847. doi:10.1214/aos/1030563990. ISSN 0090-5364. Retrieved 2023-11-16.
  3. ^ Robert, Christian P.; Casella, George (2004). "Monte Carlo Statistical Methods". Springer Texts in Statistics. doi:10.1007/978-1-4757-4145-2. ISSN 1431-875X.
  4. ^ Levin, David; Peres, Yuval; Wilmer, Elizabeth (2008-12-09). Markov Chains and Mixing Times. Providence, Rhode Island: American Mathematical Society. ISBN 978-0-8218-4739-8.
  5. ^ PESKUN, P. H. (1973). "Optimum Monte-Carlo sampling using Markov chains". Biometrika. 60 (3): 607–612. doi:10.1093/biomet/60.3.607. ISSN 0006-3444.