Subadditivity of the Partition Function (up to boundary terms)

This lemma concerns the finite-volume partition function associated with a lattice Hamiltonian and the corresponding lattice partition function .

Statement

Let $\Lambda_1,\Lambda_2\subset \mathbb{Z}^d$ be disjoint finite regions. Write the finite-volume Hamiltonian on $\Lambda_1\cup\Lambda_2$ as

where $H_{\mathrm{cross}}$ contains the interaction terms that involve sites in both $\Lambda_1$ and $\Lambda_2$.

Assume there is a constant $C\ge 0$ such that for all configurations $\sigma$,

where $|\partial(\Lambda_1,\Lambda_2)|$ counts the interaction terms (e.g. edges/plaquettes/hyperedges, depending on the model) that “cross” between $\Lambda_1$ and $\Lambda_2$.

Then for every inverse temperature $\beta\ge 0$,

Equivalently,

Key hypotheses and conclusions

Hypotheses

• $\Lambda_1,\Lambda_2$ are disjoint finite volumes.
• The interaction across the interface is uniformly bounded from below by a constant times a boundary size: $H_{\mathrm{cross}}(\sigma)\ge -C|\partial(\Lambda_1,\Lambda_2)|$. (This holds, for example, for finite-range, uniformly bounded interactions.)

Conclusions

• $\log Z_\Lambda$ is subadditive up to a boundary correction.
• For “regular” shapes (e.g. cubes), $|\partial(\Lambda_1,\Lambda_2)|$ grows like surface area, so the correction is lower order compared to volume.

Proof idea / significance

Using the decomposition $H_{\Lambda_1\cup\Lambda_2}=H_{\Lambda_1}+H_{\Lambda_2}+H_{\mathrm{cross}}$ and the bound $e^{-\beta H_{\mathrm{cross}}(\sigma)}\le e^{\beta C|\partial(\Lambda_1,\Lambda_2)|}$, one obtains $Z_{\Lambda_1\cup\Lambda_2}\le e^{\beta C|\partial|}\sum e^{-\beta H_{\Lambda_1}}e^{-\beta H_{\Lambda_2}} = e^{\beta C|\partial|}Z_{\Lambda_1}Z_{\Lambda_2}$.

This estimate is a standard input to proving existence of the thermodynamic limit of the lattice pressure (pressure ) via subadditivity arguments and Fekete's lemma , as used in existence of the thermodynamic pressure .