Critical Phenomena on the Bethe Lattice

We investigate the critical behavior of a family of $\mathbb{Z}_2$-symmetric scalar field theories on the Bethe lattice (the tree limit of regular hyperbolic tessellations) using both the non-perturbative Functional Renormalization Group and lattice perturbation theory. The family is indexed by the parameter $ζ\in (0,1]$, which determines the range of the theory via the kinetic term constructed from the graph Laplacian raised to the power $ζ$. Specifically, $ζ=1$ is the short-range theory, while $0