Two-Point Correlation Function

The ensemble expectation ⟨A(x)B(y)⟩ of observables at two points, encoding spatial (or temporal) correlations.

Two-Point Correlation Function

Many statistical-mechanical models are built from local degrees of freedom (spins, densities, fields) living on a lattice or in continuous space. For local observables $A_x$ and $B_y$ (e.g. “spin at site $x$ ”), the two-point correlation function measures how values at two locations co-vary under a chosen ensemble such as the canonical ensemble or a more general generalized Gibbs ensemble .

The (possibly uncentered) two-point correlation function is

G_{AB}(x,y) \;:=\; \langle A_x\,B_y\rangle.

If the system is translation invariant in equilibrium, then $G_{AB}(x,y)$ depends only on the separation $r=x-y$ , and one often writes

G_{AB}(r) := \langle A_0\,B_r\rangle.

This object is distinct from, but closely related to, the connected correlation function , which removes the part coming from the separate one-point means.

Connected vs. unconnected

Writing fluctuations $\delta A_x = A_x-\langle A_x\rangle$ (see fluctuation of an observable ), the connected two-point function is

G^{(c)}_{AB}(x,y) \;:=\; \langle \delta A_x\,\delta B_y\rangle \;=\; \langle A_x B_y\rangle - \langle A_x\rangle\langle B_y\rangle.

Thus $G^{(c)}_{AB}(x,y)$ is a spatial version of covariance .

Key formulas and interpretations

Independence test: If the degrees of freedom at $x$ and $y$ are effectively independent under the ensemble, then $G^{(c)}_{AB}(x,y)\approx 0$ (no genuine correlation between fluctuations).
Correlation length: In phases with short-range correlations, the connected function often decays with distance and defines a correlation length $\xi$ , heuristically via behavior like $G^{(c)}_{AB}(r)\sim e^{-|r|/\xi}$ at large $|r|$ .
Susceptibility as an integrated correlation: For many models, the linear response (a susceptibility ) can be expressed as a sum/integral of connected two-point correlations, reflecting that macroscopic response is the accumulation of microscopic correlated fluctuations.