Statistical mechanics is all about taking thermodynamic limits and, as far as I know, there are more than one way to define such limits. Consider the following theorem:

**Theorem:** In the thermodynamic limit, the pressure:
$$\psi(\beta,h) := \lim_{\Lambda \uparrow \mathbb{Z}^{d}}\psi_{\Lambda}^{\#}(\beta, h) $$
is well-defined and independent of the sequence $\Lambda \uparrow \mathbb{Z}^{d}$ and of the type of the boundary condition $\#$.

Here, I'm using the same notation and conventions from chapter 3 of Velenik and Friedli's book. The notation $\Lambda \uparrow \mathbb{Z}^{d}$ stands for the convergence in the sense of Van Hove.

**Definition** [Convergence in the sense of Van Hove] A sequence $\{\Lambda_{n}\}_{n\in \mathbb{N}}$ of (finite) subsets of $\mathbb{Z}^{d}$ is said to converge to $\mathbb{Z}^{d}$ in the sense of Van Hove if all three properties listed below are satisfied:

(1) $\{\Lambda_{n}\}_{n\in \mathbb{N}}$ is an *increasing* sequence of subsets.

(2) $\bigcup_{n\in \mathbb{N}}\Lambda_{n} = \mathbb{Z}^{d}$

(3) $\lim_{n\to \infty}\frac{|\partial^{in}\Lambda_{n}|}{|\Lambda_{n}|} = 0$, where $|X|$ denotes the cardinality of the set $X$ and $\partial^{in}\Lambda:=\{i\in \Lambda: \hspace{0.1cm} \exists j \in\Lambda^{c} \hspace{0.1cm} \mbox{with} \hspace{0.1cm} |i-j|=1 \}$

My point here is the following. Convergence in the sense of Van Hove is a notion of convergence of *sets*, not *functions of sets*. But what does $\lim_{\Lambda\uparrow \mathbb{Z}^{d}}\psi^{\#}_{\Lambda}(\beta, h)$ mean?

areknown to depend on the choice of boundary conditions, e.g the free energy of the six-vertex model (which has d=2) with periodic vs domain-wall boundaries. I believe that the conditions of the definition are met; the point is that the local (divergencelessness) constraint of the model causes the choice of boundary conditions to influence a macroscopic portion of the lattice. $\endgroup$ – Jules Lamers Jul 22 '20 at 9:54