Consider the 1-dimensional Ising model $$\mathcal{H}_N=-J\sum_{i=1}^{N} \sigma_i\sigma_{i+1}-H\sum_{i=1}^{N}\sigma_i\qquad\sigma_i=\pm 1$$

with periodic boundary conditions $\sigma_{N+1}=\sigma_1$ and magnetic field $H$. Let $\Omega$ be the set of the one-sided infinite sequences $\omega=(\sigma_1,\sigma_2,\dots)\in\Omega$. Consider the map $x:\Omega\mapsto\{-1,1\}$, with coordinate function $x_n(\omega)=\sigma_n$ and the shift transformation $T:\Omega\mapsto\Omega$, with $\,x_n(T\omega)=x_{n+1}(\omega)=\sigma_{n+1}$. Let further ${\cal F}$ be a $\sigma$-algebra of subsets of $\Omega$, which is generated by the sets (cylinders) of the form $\{\omega\in\Omega:(\sigma_i,\dots,\sigma_{i+n-1})\in E\},$ while $E$ is a subset of $\{-1,1\}^n$. The $\sigma$-fields are generated by what is called "thin" cylinders, that is, sets of the form $\{\omega\in\Omega: x_l(\omega)=\sigma_l, i\le l\le i+n-1\}$

Since the finite disjoint unions of thin cylinders form a field which generates ${\cal F}$,

a measure $P$ on ${\cal F}$ is uniquely determined by the values

$$p_n(s_i,\dots,s_{i+n-1})=P\{\omega\in\Omega:x_i(\omega)=s_i,\dots, x_{i+n-1}(\omega)=s_{i+n-1}\}$$

for $i\ge 1$. Now, for any finite $n, N$, $1\le n\le N$ and $1\le i\le N-n+1$, one can define the partial trace:

$$Z_{N}(i,n):=\sum_{\substack{\sigma_1,\dots,\sigma_{i-1},\\ \sigma_{i+n},\dots,\sigma_{N}}}

e^{-\beta\mathcal{H}(\sigma_1,\dots,\sigma_{N})}$$

and the ordinary trace:

$$Z_{N}:=\sum_{\sigma_1,\dots,\sigma_N} e^{-\beta\mathcal{H}(\sigma_1,\dots,\sigma_N)}.$$

Then, consider the fraction $$p^{(N)}_n(s_i,\dots,s_{i+n-1})=\frac{Z_{N}(i,n)}{Z_N}.$$

From the latter, we can derive a probability measure in the following sense

$$p_n(s_i,\dots,s_{i+n-1})=\lim_{N\rightarrow\infty}p^{(N)}_n(s_i,\dots,s_{i+n-1}),$$

and the associated transition probability is simply

$$p(s_1,\dots,s_n|s_{n+1})=\frac{p_{n+1}(s_1,\dots,s_{n+1})}{p_n(s_1,\dots,s_n)}.$$

By substitution of the traces and some further algebraic manipulations one can show that the latter is equal to a stochastic and irreducible transition matrix. This implies that the transition probability is associated with an ergodic Markov shift.