Anomalous criticality coexists with giant cluster in the uniform forest model

In percolation theory, the general scenario for the supercritical phase is that all clusters, except the unique giant one, are small and the two-point correlation exponentially decays to some constant. We show by extensive simulations that the whole supercritical phase of the three-dimensional uniform forest model simultaneously exhibits an infinite tree and a rich variety of critical phenomena. Besides typical scalings like algebraically decaying correlation, power-law distribution of cluster sizes, and divergent correlation length, a number of anomalous behaviors emerge. The fractal dimensions for off-giant trees take different values when being measured by linear system size or gyration radius. The giant tree size displays two-length scaling fluctuations, instead of following the central-limit theorem. In a non-Gaussian fermionic field theory, these unusual properties are closely related to the non-abelian continuous OSP$(1|2)$ supersymmetry in the fermionic hyperbolic plane ${\mathbb H}^{0|2}$.