Griffiths monotonicity lemma

Statement (monotonicity in parameters)

Consider the ferromagnetic Ising model on a finite graph/finite region $\Lambda$ with spins $\sigma_x\in\{-1,+1\}$ and Hamiltonian

with ferromagnetic couplings $J_{xy}\ge 0$ and nonnegative external field $h_x\ge 0$ (and any fixed boundary condition, typically plus).

Let $\langle \cdot \rangle_{J,h}$ denote expectation with respect to the associated finite-volume Gibbs measure .

For any finite set $A\subset \Lambda$, define the spin product $\sigma_A := \prod_{x\in A}\sigma_x$. Then:

• For every edge $\{x,y\}$,

  $$\frac{\partial}{\partial J_{xy}} \langle \sigma_A \rangle_{J,h} \ge 0.$$

• For every site $x$,

  $$\frac{\partial}{\partial h_x} \langle \sigma_A \rangle_{J,h} \ge 0.$$

Equivalently, all correlation functions $\langle \sigma_A\rangle_{J,h}$ are nondecreasing functions of each coupling $J_{xy}$ and each field $h_x$ in the ferromagnetic, $h\ge 0$ regime.

Key hypotheses

• The model is ferromagnetic: $J_{xy}\ge 0$ for all interacting pairs.
• External field is nonnegative: $h_x\ge 0$ (a common hypothesis ensuring the relevant correlation inequalities hold in the strongest form).
• Differentiability of finite-volume expectations in the parameters (automatic for finite systems).

Key conclusions

• Monotonicity of magnetization $m_x=\langle \sigma_x\rangle$ and higher correlations with respect to strengthening couplings/fields.
• Useful monotone limits as volume grows (e.g., with plus boundary conditions), supporting construction of infinite-volume Gibbs measures by monotone convergence.
• Parameter comparison and bounds for observables such as two-point correlations .

Cross-links to relevant definitions and tools

• The model: Ising model .
• Expectation as an integral w.r.t. a probability measure: expectation .
• Positivity inequalities typically used in the proof:
  • Griffiths inequalities
  • GKS inequalities
  • FKG inequality

Proof idea (sketch)

Differentiate the expectation in the parameter. For example,

Similarly, $$ \frac{\partial}{\partial J_{xy}}\langle \sigma_A\rangle_{J,h}

={} \operatorname{Cov}_{J,h}(\sigma_A,\sigma_x\sigma_y). $$

In the ferromagnetic regime with $h\ge 0$, Griffiths/GKS/FKG-type inequalities imply the relevant covariances are nonnegative, giving the stated monotonicity.