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In percolation theory, the general scenario for the supercritical phase isthat all clusters, except the unique giant one, are small and the two-pointcorrelation exponentially decays to some constant. We show by extensivesimulations that the whole supercritical phase of the three-dimensional uniformforest model simultaneously exhibits an infinite tree and a rich variety ofcritical phenomena. Besides typical scalings like algebraically decayingcorrelation, power-law distribution of cluster sizes, and divergent correlationlength, a number of anomalous behaviors emerge. The fractal dimensions foroff-giant trees take different values when being measured by linear system sizeor gyration radius. The giant tree size displays two-length scalingfluctuations, instead of following the central-limit theorem. In a non-Gaussianfermionic field theory, these unusual properties are closely related to thenon-abelian continuous OSP$(1|2)$ supersymmetry in the fermionic hyperbolicplane ${\mathbb H}^{0|2}$.

Anomalous criticality coexists with giant cluster in the uniform forest model