= One-dimensional Ising correlation function
{c}
For the zero-field nearest-neighbour <Ising model> on $\mathbb Z$, take
$$
H=-J\sum_n\sigma_n\sigma_{n+1},\qquad
\sigma_n=\pm1,\qquad \beta=\frac1{k_BT},\quad T>0.
$$
The infinite-chain <correlation function> is
$$
\langle\sigma_x\sigma_y\rangle
=\left[\tanh(\beta J)\right]^{|x-y|}.
$$
Writing $\tanh\beta$ assumes the <energy> unit $J=1$; with the parent's dimensional coupling, the factor $J$ must remain.
The zero-field <Boltzmann distribution> makes consecutive <spins> a stationary <Markov chain> with
$$
P(\sigma_{n+1}=s'\mid\sigma_n=s)
=\frac{e^{\beta Jss'}}{2\cosh(\beta J)}.
$$
Hence $\mathbb E[\sigma_{n+1}\mid\sigma_n]=\sigma_n\tanh(\beta J)$. Iterating and using $\sigma_x^2=1$ proves the two-point formula.
For ordered sites $x_1<\cdots<x_{2m}$, the same conditional-expectation argument gives
$$
\langle\sigma_{x_1}\cdots\sigma_{x_{2m}}\rangle
=\left[\tanh(\beta J)\right]^{\sum_{r=1}^m(x_{2r}-x_{2r-1})}.
$$
Odd products vanish by <spin inversion symmetry>. Positive $J$ gives positive correlations; negative $J$ gives alternating signs. For nonzero $J$, their magnitude decays with <correlation length> $\xi=-1/\log|\tanh(\beta J)|$.
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