Regularity of Gibbs measures for unbounded spin systems on general graphs

We consider a general class of spin systems with potentially unbounded real-valued spins, defined via a single-site potential with super-Gaussian tails on general graphs, allowing for both short- and long-range interactions. This class includes all $P(\varphi)$ models, in particular the well-studied $\varphi^4$ model. We construct an infinite-volume extremal measure called the plus measure as the limit of finite-volume Gibbs measures with weakly growing boundary conditions and show that it is regular, in the sense that it admits a bounded Radon-Nikodym derivative with respect to a product measure of single-site distributions with super-Gaussian tails. Moreover, we provide an alternative construction of the plus measure as the limit of finite-volume Gibbs measures that are regular up to the boundary. As a key intermediate step, we establish regularity and tightness of finite-volume Gibbs measures for a large class of growing boundary conditions $\xi$. Our regularity estimates are encoded in terms of a function $A(\xi)$, which provides precise control on the change of measure induced by boundary perturbations, and can thus be viewed as an analogue of the Cameron-Martin theorem for non-Gaussian fields. In the nearest-neighbour case, this class includes boundary conditions that grow at most double-exponentially in the distance to the boundary when the single-site measure has tails of the form $e^{-a|u|^n}$ for some $n>2$.Our results apply to arbitrary graphs and improve upon earlier results of Lebowitz and Presutti, and Ruelle, which apply in the context of $\mathbb{Z}^d$ and allow only logarithmically growing boundary conditions, as well as subsequent extensions to vertex-transitive graphs of polynomial growth.