Connected correlations as derivatives of expectations

Let $\mu_\lambda$ be a Gibbs/ensemble measure (e.g. canonical ensemble or finite-volume Gibbs measure ) whose Hamiltonian depends on a real parameter $\lambda$ via a linear coupling to an observable $A$:

Let $\langle \cdot\rangle_\lambda$ denote ensemble averages with respect to $\mu_\lambda$.

Statement

For any integrable observable $B$,

In other words,

where $\mathrm{Cov}_\lambda(A,B)$ is the (two-point) connected correlation, i.e. the connected two-point function at the level of observables.

In particular, for the partition function $Z(\lambda)$ (canonical or lattice ),

with variance as in variance in an ensemble .

More generally, if one introduces sources $\boldsymbol{\lambda}=(\lambda_1,\dots,\lambda_k)$ coupled to observables $(A_1,\dots,A_k)$, then mixed partial derivatives of $\log Z(\boldsymbol{\lambda})$ generate higher connected correlations (cumulants).

Key hypotheses

• The parameter enters linearly: $H_\lambda=H_0-\lambda A$ (or, more generally, differentiably with $\partial_\lambda H_\lambda=-A$).
• The relevant derivatives can be passed under the integral/sum defining $\langle\cdot\rangle_\lambda$ (automatic in finite volume with bounded observables).
• $\beta>0$ is fixed (inverse temperature).

Conclusions

• Response of $B$ to the field conjugate to $A$ is controlled by their covariance.
• Taking $B=A$ recovers a fluctuation identity: $\partial_\lambda\langle A\rangle_\lambda=\beta\,\mathrm{Var}_\lambda(A)\ge 0.$
• This packages many “susceptibility = fluctuation” statements, including susceptibility–variance for magnetization as a special case.