Connected Correlation Function

Correlation functions measure how observables fluctuate together in an ensemble. The connected correlation function subtracts the part explained by separate averages, isolating the genuinely correlated component. For two observables, this is exactly a covariance; for higher orders, it generalizes to cumulants.

Two-point definition

For observables $A$ and $B$,

Equivalently, in terms of the fluctuations $\delta A = A-\langle A\rangle$ and $\delta B = B-\langle B\rangle$,

For local observables $A_x,B_y$, the connected two-point function is

and it is the connected version of the two-point correlation function .

Relation to variance and covariance

• Setting $A=B$ gives the variance:

  $$\langle A^2\rangle_c = \mathrm{Var}(A),$$

  which connects connectedness to ensemble variance .

• In general,

  $$\langle AB\rangle_c = \mathrm{Cov}(A,B),$$

  matching ensemble covariance .

Connected correlations as cumulants

In Gibbs-type ensembles, connected correlations are the natural objects produced by differentiating $\log Z$ rather than $Z$. This is the content of connected correlations as cumulants and cumulant generating functions : derivatives of $\log Z$ with respect to fields coupled to observables generate connected correlation functions.

Physical interpretation

• Remove “trivial” correlations: If $\langle A\rangle$ and $\langle B\rangle$ are nonzero, the unconnected correlator $\langle AB\rangle$ can be dominated by the product $\langle A\rangle\langle B\rangle$. The connected correlator subtracts this, leaving the part due to correlated fluctuations.

• Clustering and phases: In many equilibrium phases away from criticality, connected correlations decay with separation, defining a finite correlation length . Near critical points, connected correlations can become long-ranged, and their spatial integral controls macroscopic responses such as susceptibility .