Consider a statistical mechanical system (say the 1D Ising model) on a finite lattice of size $N$, and call the corresponding partition function (as a function of, say, real temperature and real magnetic field) $Z^{(N)}(t, h)$, where $t$ is temperature and $h$ - the magnetic field.

The partition function $Z$ is analytic (in finite volume $N$) and doesn't admit any zeros. However, as soon as one passes to complex field $h$ (or temperature, but let's consider complex field here), $Z$ admits zeros on the unit circle $S^1$ in $\mathbb{C}$.

Call the set of zeros $\mathcal{Z}_N$, where $N$ emphasizes finite lattice of size $N$. It is in general a nontrivial problem to decide whether the sequence of sets $\{\mathcal{Z}_N\}_{N\in\mathbb{N}}$ accumulates on some set in $S^1$, and if it does, to describe the topology of this limit set, which we'll call $\mathcal{Z}$.

Now, suppose that for a given system we proved that there does indeed exist a nonempty set $\mathcal{Z}$ such that $\mathcal{Z}_N\rightarrow\mathcal{Z}$ as $N\rightarrow\infty$ (in some sense - say in Hausdorff metric).

Is there a natural measure $\mu$ defined on $\mathcal{Z}$ that has physical meaning? If so, what sort of properties of this measure are physically relevant (say, relating to phase transitions)?

In my mind this is quite a natural question, because it translates into "Is there a natural way to measure the set where the system develops critical behavior?"

For example, one candidate would be the Hausdorff dimension. But I am interested more in something that would measure the density of zeros in a natural way (such as, for example, the density of states measure for quantum Hamiltonians).

**EDIT:** I know, of course, that the 1D Ising model is exactly solvable when interaction strength and magnetic field are constant. Here I implicitly assume that interaction (nearest neighbor, to keep it simple) and/or the magnetic field depend on lattice sites.

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