One-dimensional Ising correlation function

For the zero-field nearest-neighbour Ising model on $\mathbb{Z}$, takeThe infinite-chain correlation function isWriting $\tanh \beta$ 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 withHence $\mathbb{E}[{\sigma }_{n+1}\mid {\sigma }_n]={\sigma }_n\tanh (\beta J)$. 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 givesOdd 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 $\xi =-1/\log |\tanh (\beta J)|$.