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PreprintCC BY 4.0arXiv

Iterated Graph Systems (III): Alexander-Orbach conjecture on hierarchical lattices

  • Ziyu Neroli
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Abstract

Can the Alexander-Orbach conjecture hold on hierarchical lattices? We construct Gromov-Hausdorff-Prokhorov scaling limits of critical percolation clusters on hierarchical lattices in the chemical metric and determine the typical spectral dimensions of the associated diffusions. These dimensions are dense in (1,2); however, outside a narrow region in dimension space, we prove that the Alexander-Orbach conjecture fails on every hierarchical lattice under consideration.

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Cite

BibTeXRIS
BibTeX
@misc{neroli2026iterated,
  title = {{Iterated Graph Systems (III): Alexander-Orbach conjecture on hierarchical lattices}},
  author = {Ziyu Neroli},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.16875},
  eprint = {2610.16875},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.16875},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Iterated Graph Systems (III): Alexander-Orbach conjecture on hierarchical lattices
AU  - Ziyu Neroli
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2610.16875
UR  - https://arxiv.org/abs/2610.16875
AB  - Can the Alexander-Orbach conjecture hold on hierarchical lattices? We construct Gromov-Hausdorff-Prokhorov scaling limits of critical percolation clusters on hierarchical lattices in the chemical metric and determine the typical spectral dimensions of the associated diffusions. These dimensions are dense in (1,2); however, outside a narrow region in dimension space, we prove that the Alexander-Orbach conjecture fails on every hierarchical lattice under consideration.
ER  -