Construction of an irreducible Markov chain in the Ising model について

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Construction of an irreducible Markov chain in the Ising model ：ウィキペディア英語版

Construction of an irreducible Markov chain in the Ising model

In applied mathematics, construction of an irreducible Markov Chain in the Ising model is the first step in overcoming a computational obstruction encountered when a Markov chain Monte Carlo method is used to get an exact goodness-of-fit test for the finite Ising model.
The Ising model was used to study magnetic phase transitions at the very beginning, and now it is one of the most famous models of interacting systems.
== Markov bases ==
For every integer vector

z\in Z^

, we can uniquely write it as

z=z^+-z^-

, where

z^+

and

z^-

are nonnegative vectors. Then the Markov basis in Ising model can be degined as:
A Markov bases for the Ising model is a set

\widetilde\subset Z ^

os integer vector such that:
(i) For all

z\in \widetilde

there must be

T_1(z^+)=T_1(z^-)

and

T_2(z^+)=T_2(z^-)

.
(ii) For any

a,b\in Z_

and any

x,y\in S(a,b)

there always exist

z_1,\ldots,z_k \in \widetilde

satisfy
:

y=x+\sum_^k z_i

and
:

x+\sum_^l z_i\in S(a,b)

for ''l'' = 1,...,''k''.
The element of

\widetilde

is move. Then using the Metropolis–Hastings algorithm, we can get an aperiodic, reversible and irreducible Markov Chain.
The paper published by P.DIACONIS AND B.STURMFELS in 1998 ‘Algebraic algorithms for sampling from conditional distributions’ shows that a Markov basis can be defined algebraically as in Ising model
Then by the paper published by P.DIACONIS AND B.STURMFELS in 1998, any generating set for the ideal

I:=\ker(*)

is a Markov basis for the Ising model.

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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