Alexander–Orbach conjecture on universal spectral dimension at percolation criticality

Prove that, at the percolation threshold in isotropic percolation, the giant connected component has a universal spectral dimension ds = 4/3 independent of spatial dimension.

Background

In analyzing link dilution on lattices, the authors relate the emergence of a random-tree-like structure at the percolation threshold to the Alexander–Orbach conjecture. This conjecture provides a universal prediction for the spectral dimension at criticality across dimensions and motivates the interpretation of their numerical findings.

Establishing this result rigorously is fundamental for connecting percolation geometry to dynamical properties governed by the Laplacian spectrum.

References

This is a direct consequence of the Alexander-Orbach conjecture , which predicts that, at criticality, the giant component for isotropic percolation has a universal spectral dimension $d_s=\nicefrac{4}{3}$ across all dimensions.

— Geometric Criticality in Scale-Invariant Networks (2507.11348 - Lucarini et al., 15 Jul 2025) in Structural sparsity section, paragraph discussing percolation at critical dilution