Critical Phenomena on the Bethe Lattice

We investigate the critical behavior of a family of Z2-symmetric scalar field theories on the Bethe lattice (the tree limit of regular hyperbolic tessellations) using both the non-perturbative Functional Renormalization Group and perturbation theory. Due to the hyperbolic nature of Bethe lattices, the Laplacian lacks a zero mode and exhibits a spectral gap. We demonstrate that closing the spectral gap via a modified Laplacian leads to novel critical behavior governed by interacting fixed points. This stands in contrast to the nearest-neighbor Ising model, which exhibits a phase transition with mean-field critical exponents. We further comment on the possible reasons for such a deviation.