Gibbsian structure in random point fields has been a classical tool for
studying their spatial properties. However, exact Gibbs property is available
only in a relatively limited class of models, and it does not adequately
address many random fields with a strongly dependent spatial structure. In this
work, we provide a general framework for approximate Gibbsian structure for
strongly correlated random point fields. These include processes that exhibit
strong spatial rigidity, in particular, a certain one-parameter family of
analytic Gaussian zero point fields, namely the $\alpha$-GAFs. Our framework
entails conditions that may be verified via finite particle approximations to
the process, a phenomenon that we call an approximate Gibbs property. We show
that these enable one to compare the spatial conditional measures in the
infinite volume limit with Gibbs-type densities supported on appropriate
singular manifolds, a phenomenon we refer to as a generalized Gibbs property.
We demonstrate the scope of our approach by showing that a generalized Gibbs
property holds with a logarithmic pair potential for the $\alpha$-GAFs for any
value of $\alpha$. This establishes the level of rigidity of the $\alpha$-GAF
zero process to be exactly $\lfloor \frac{1}{\alpha} \rfloor$, settling in the
affirmative an open question regarding the existence of point processes with
any specified level of rigidity. For processes such as the zeros of
$\alpha$-GAFs, which involve complex, many-body interactions, our results imply
that the local behaviour of the random points still exhibits 2D Coulomb-type
repulsion in the short range. Our techniques can be leveraged to estimate the
relative energies of configurations under local perturbations, with possible
implications for dynamics and stochastic geometry on strongly correlated random
point fields.
