Construction of an irreducible Markov chain in the Ising model

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Construction of an irreducible Markov chain in the Ising model is a mathematical method to prove results.

In applied

The Ising model is used to study magnetic phase transitions and is one of the models of interacting systems.^[1]

Markov bases

Every integer vector ${\displaystyle z\in Z^{N_{1}\times \cdots \times N_{d}}}$, can be uniquely written as ${\displaystyle z=z^{+}-z^{-}}$, where ${\displaystyle z^{+}}$ and ${\displaystyle z^{-}}$ are non-negative vectors. A Markov basis for the Ising model is a set ${\displaystyle {\widetilde {Z}}\subset Z^{N_{1}\times \cdots \times N_{d}}}$ of integer vectors such that:

(i) For all ${\displaystyle z\in {\widetilde {Z}}}$, there must be ${\displaystyle T_{1}(z^{+})=T_{1}(z^{-})}$ and ${\displaystyle T_{2}(z^{+})=T_{2}(z^{-})}$.

(ii) For any ${\displaystyle a,b\in Z_{>0}}$ and any ${\displaystyle x,y\in S(a,b)}$, there always exist ${\displaystyle z_{1},\ldots ,z_{k}\in {\widetilde {Z}}}$ satisfy:

${\displaystyle y=x+\sum _{i=1}^{k}z_{i}}$

and

${\displaystyle x+\sum _{i=1}^{l}z_{i}\in S(a,b)}$

for l=1,...,k.

The element of ${\displaystyle {\widetilde {Z}}}$ is moved. Then, by using the

The paper "Algebraic algorithms for sampling from conditional distributions", published by Persi Diaconis and Bernd Sturmfels in 1998,^[2] shows that a Markov basis can be defined algebraically as an Ising model. According to the paper, any generating set for the ideal ${\displaystyle I:=\ker({\psi }*{\phi })}$ is a Markov basis for the Ising model.

Construction of an irreducible Markov Chain

Without modifying the algorithm mentioned in the paper, it is impossible to get uniform samples from ${\displaystyle S(a,b)}$, otherwise leading to inaccurate p-values.^[3]

A simple swap is defined as ${\displaystyle z\in Z^{N_{1}\times \cdots \times N_{d}}}$ of the form ${\displaystyle z=e_{i}-e_{j}}$, where ${\displaystyle e_{i}}$ is the canonical basis vector of ${\displaystyle z\in Z^{N_{1}\times \cdots \times N_{d}}}$. Simple swaps change the states of two lattice points in y.

Z denotes the set of sample swaps. Then two configurations ${\displaystyle y',y\in S(a,b)}$ are ${\displaystyle S(a,b)}$-connected by Z, if there is a path between ${\displaystyle y}$ and ${\displaystyle y'}$ in ${\displaystyle S(a,b)}$ consisting of simple swaps ${\displaystyle z\in Z}$, which means there exists ${\displaystyle z_{1},\ldots ,z_{k}\in Z}$ such that:

${\displaystyle y'=y+\sum _{i=1}^{k}z_{i}}$

with

${\displaystyle y+\sum _{i=1}^{l}z_{i}\in S(a,b)}$

for ${\displaystyle l=1\ldots k}$

The algorithm can now be described as:

(i) Start with the Markov chain in a configuration ${\displaystyle y\in S(a,b)}$

(ii) Select ${\displaystyle z\in Z}$ uniformly at random and let ${\displaystyle y'=y+z}$.

(iii) Accept ${\displaystyle y'}$ if ${\displaystyle y'\in S(a,b)}$; 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, we can overcome this problem by using the Metropolis-Hastings algorithm in the smallest expanded sample space ${\displaystyle S^{\star }(a,b)}$.

Irreducibility in the 1-dimensional Ising model

The proof of