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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    $\begingroup$ I think the original paper by Yang and Lee explains this. I believe they actually show (or just assume) that in the thermodynamic limit there is a well-behaved density of zeros, and then use this density to calculate things like critical exponents. $\endgroup$ – genneth Mar 2 '12 at 18:46
  • $\begingroup$ I of course suspected that it should be (naturally) zero distribution. In fact, I think the distribution of zeros measure in the thermodynamic limit is just weak limit of distribution of zeros measures on finite lattices (as the lattice size grows to infinity). Thank you for the reference! $\endgroup$ – user3657 Mar 2 '12 at 19:41
  • $\begingroup$ @genneth: you're right. Note, however, that, as far as I know, one can say essentially nothing about the limiting density (in dimensions 2 and higher, at low temperature). It is, e.g., not known whether the distribution of zeros in the thermodynamic limit allows analytic continuation from {Re(h)>0} to {Re(h)<0} (it is only known that such an analytic continuation cannot be done through h=0, since there is an essential singularity there). $\endgroup$ – Yvan Velenik Mar 2 '12 at 20:11
  • $\begingroup$ @Yvan: Also, in dimension one, as far as I know, depending on choice of interaction, distribution of zeros may also present a nontrivial problem, or am I wrong? In other words, I think even in dimension one, results and techniques are model dependent. Is this true? $\endgroup$ – user3657 Mar 2 '12 at 21:07
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    $\begingroup$ @WNY: I guess that it depends what information you want to extract (obviously, if the coupling constants are uniformly bounded and decay fast enough with the distance between the corresponding spins, then there won't be a phase transition in the model). Now, independently of what you want to extract, I'd guess that the determination of the asymptotic locations of zeros is certainly difficult (and usually impossible), in general, when the interaction is not finite-range and periodic... $\endgroup$ – Yvan Velenik Mar 3 '12 at 10:59

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