Renormalization Group Flows of Spectral Dimensions on Hierarchical Lattices: Critical Percolation, Resistance Exponents, and Transport Universality

A recurring idea in discrete approaches to quantum gravity and in network models of growth is that coarse-graining drives a graph spontaneously to a stable non-integer spectral dimension. We test this idea on bond percolation on the diamond hierarchical lattice (DHL), where the renormalization group (RG) is exact. The edge-retention probability obeys p' = 2p² − p⁴. The fixed points p = 0 (fragmented, d_s → 0) and p = 1 (intact lattice, d_s = 2) attract, and p_c = (√5 − 1)/2 repels, with ν = ln 2 / ln(6 − 2√5) ≈ 1.6353. A non-integer spectral dimension therefore appears only at a tuned, unstable critical point. The change from integer to non-integer behaviour is a crossover, not a jump. For the critical spanning cluster, including all dangling parts, the conditional moments of the cluster mass obey an exact 2×2 linear recursion. Its Perron eigenvalue Λ = 7 − 2√5 + √(73 − 32√5) gives the closed-form mass dimension d_f = log₂ Λ = 1.89929, which population dynamics reproduces to 0.1%. The pole-to-pole resistance grows with Lyapunov exponent ζ = 0.8127 ± 0.0002, and the Einstein relation gives d_s = 2d_f/(d_f + ζ) ≈ 1.401. Exact diagonalisation of generation-6 and generation-7 clusters, averaged over whole log-periods, gives 1.41 and 1.44, and annealed and quenched averages agree to 0.002. The same procedure gives 1.994 for the intact lattice, whose exact value is 2. Newman–Watts shortcuts are a relevant perturbation that raises d_s beyond τ ~ ξ² without creating a plateau. The DHL critical cluster nearly shares the mass dimension of two-dimensional percolation (1.8993 vs 91/48), but its resistance exponent (0.813 vs 0.983) and spectral dimension (≈1.40 vs 1.317) differ. Geometric similarity therefore does not imply a common transport universality class. The spectral dimension of "the" critical cluster also depends on the chosen subgraph and Laplacian: the Hambly–Kumagai diffusion on the skeleton gives about 1.68. None of these values equals 4/3 or the sometimes-posited 1.25. Code, data and figures: https://github.com/Ruqing1963/rg-spectral-dimension