@article{BenjaminiSchramm1996,
  author = {Benjamini, Itai and Schramm, Oded},
  title = {Percolation beyond {$\mathbb Z^d$}, many questions and a few answers},
  journal = {Electronic Communications in Probability},
  volume = {1},
  pages = {71--82},
  year = {1996},
  doi = {10.1214/ECP.v1-978},
  url = {https://emis.de/ft/41630}
}

@article{Hutchcroft2016,
  author = {Hutchcroft, Tom},
  title = {Critical percolation on any quasi-transitive graph of exponential growth has no infinite clusters},
  journal = {Comptes Rendus. Math\'ematique},
  volume = {354},
  number = {9},
  pages = {944--947},
  year = {2016},
  doi = {10.1016/j.crma.2016.07.013},
  eprint = {1605.05301},
  archivePrefix = {arXiv},
  url = {https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.07.013/}
}

@article{HermonHutchcroft2021,
  author = {Hermon, Jonathan and Hutchcroft, Tom},
  title = {No percolation at criticality on certain groups of intermediate growth},
  journal = {International Mathematics Research Notices},
  volume = {2021},
  number = {22},
  pages = {17433--17455},
  year = {2021},
  doi = {10.1093/imrn/rnz265},
  eprint = {1809.11112},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/1809.11112v2}
}

@article{TesseraTointon2021,
  author = {Tessera, Romain and Tointon, Matthew C. H.},
  title = {A finitary structure theorem for vertex-transitive graphs of polynomial growth},
  journal = {Combinatorica},
  volume = {41},
  pages = {263--298},
  year = {2021},
  doi = {10.1007/s00493-020-4295-6},
  eprint = {1908.06044},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/1908.06044v3}
}

@misc{KozmaNitzan2024,
  author = {Kozma, Gady and Nitzan, Shahaf},
  title = {A reduction of the {$\theta(p_c)=0$} problem to a conjectured inequality},
  year = {2024},
  eprint = {2401.12397},
  archivePrefix = {arXiv},
  primaryClass = {math.PR},
  doi = {10.48550/arXiv.2401.12397},
  url = {https://arxiv.org/abs/2401.12397v1}
}

@misc{Leder2026,
  author = {Leder, Justin},
  title = {{$\theta(p_c)=0$} for {Bernoulli} bond percolation on {$\mathbb Z^d$} in all dimensions {$d\ge2$}: A guide to the {Lean} formalization},
  year = {2026},
  month = aug,
  howpublished = {Public research announcement},
  url = {https://github.com/anthropics/formal-math/blob/795efb86f191735c5481675763537cfb4ff37e55/percolation/summary.pdf}
}

@article{VanDenBergHaggstromKahn2006,
  author = {van den Berg, J. and H{\"a}ggstr{\"o}m, O. and Kahn, J.},
  title = {Some conditional correlation inequalities for percolation and related processes},
  journal = {Random Structures \& Algorithms},
  volume = {29},
  pages = {417--435},
  year = {2006},
  doi = {10.1002/rsa.20102},
  eprint = {math/0408176},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/math/0408176}
}

@article{Hutchcroft2020Locality,
  author = {Hutchcroft, Tom},
  title = {Locality of the critical probability for transitive graphs of exponential growth},
  journal = {The Annals of Probability},
  volume = {48},
  number = {3},
  pages = {1352--1371},
  year = {2020},
  doi = {10.1214/19-AOP1395},
  eprint = {1808.08940},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/1808.08940v2}
}

@article{AizenmanKestenNewman1987,
  author = {Aizenman, Michael and Kesten, Harry and Newman, Charles M.},
  title = {Uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation},
  journal = {Communications in Mathematical Physics},
  volume = {111},
  number = {4},
  pages = {505--531},
  year = {1987},
  doi = {10.1007/BF01219071},
  url = {https://link.springer.com/article/10.1007/BF01219071}
}

@article{Harris1960,
  author = {Harris, Theodore E.},
  title = {A lower bound for the critical probability in a certain percolation process},
  journal = {Proceedings of the Cambridge Philosophical Society},
  volume = {56},
  number = {1},
  pages = {13--20},
  year = {1960},
  doi = {10.1017/S0305004100034241}
}

@article{Kesten1980,
  author = {Kesten, Harry},
  title = {The critical probability of bond percolation on the square lattice equals {$1/2$}},
  journal = {Communications in Mathematical Physics},
  volume = {74},
  number = {1},
  pages = {41--59},
  year = {1980},
  doi = {10.1007/BF01197577}
}

@article{HaraSlade1990,
  author = {Hara, Takashi and Slade, Gordon},
  title = {Mean-field critical behaviour for percolation in high dimensions},
  journal = {Communications in Mathematical Physics},
  volume = {128},
  number = {2},
  pages = {333--391},
  year = {1990},
  doi = {10.1007/BF02108785}
}

@article{FitznerHofstad2017,
  author = {Fitzner, Robert and van der Hofstad, Remco},
  title = {Mean-field behavior for nearest-neighbor percolation in {$d>10$}},
  journal = {Electronic Journal of Probability},
  volume = {22},
  pages = {1--65},
  note = {Paper No. 43},
  year = {2017},
  doi = {10.1214/17-EJP56},
  eprint = {1506.07977},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/1506.07977}
}

@book{Robinson1996,
  author = {Robinson, Derek J. S.},
  title = {A Course in the Theory of Groups},
  edition = {2},
  series = {Graduate Texts in Mathematics},
  volume = {80},
  publisher = {Springer},
  address = {New York},
  year = {1996},
  doi = {10.1007/978-1-4419-8594-1}
}

@article{VanDenBergKahn2001,
  author = {van den Berg, J. and Kahn, J.},
  title = {A correlation inequality for connection events in percolation},
  journal = {The Annals of Probability},
  volume = {29},
  number = {1},
  pages = {123--126},
  year = {2001},
  doi = {10.1214/aop/1008956324},
  url = {https://doi.org/10.1214/aop/1008956324}
}

@article{Gromov1981,
  author = {Gromov, Mikhael},
  title = {Groups of polynomial growth and expanding maps},
  journal = {Publications Math\'ematiques de l'IH\'ES},
  volume = {53}, pages = {53--78}, year = {1981},
  note = {With an appendix by Jacques Tits},
  doi = {10.1007/BF02698687},
  url = {https://www.numdam.org/item/PMIHES_1981__53__53_0/}
}

@article{Trofimov1985,
  author = {Trofimov, V. I.},
  title = {Graphs with polynomial growth},
  journal = {Mathematics of the USSR-Sbornik},
  volume = {51}, number = {2}, pages = {405--417}, year = {1985},
  note = {English translation of Matematicheskii Sbornik 123(165) (1984), 407--421},
  doi = {10.1070/SM1985v051n02ABEH002866},
  url = {https://www.mathnet.ru/eng/sm2028}
}

@article{Trofimov2003,
  author = {Trofimov, V. I.},
  title = {Undirected and directed graphs with near polynomial growth},
  journal = {Discussiones Mathematicae Graph Theory},
  volume = {23}, number = {2}, pages = {383--391}, year = {2003},
  doi = {10.7151/dmgt.1208},
  url = {https://eudml.org/doc/270249}
}

@article{BreuillardGreenTao2012,
  author = {Breuillard, Emmanuel and Green, Ben and Tao, Terence},
  title = {The structure of approximate groups},
  journal = {Publications Math\'ematiques de l'IH\'ES},
  volume = {116}, pages = {115--221}, year = {2012},
  doi = {10.1007/s10240-012-0043-9},
  eprint = {1110.5008}, archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/1110.5008}
}

@article{GrimmettMarstrand1990,
  author = {Grimmett, G. R. and Marstrand, J. M.},
  title = {The supercritical phase of percolation is well behaved},
  journal = {Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences},
  volume = {430},
  number = {1879},
  pages = {439--457},
  year = {1990},
  doi = {10.1098/rspa.1990.0100},
  url = {https://doi.org/10.1098/rspa.1990.0100}
}

@article{MartineauTassion2017,
  author = {Martineau, S{\'e}bastien and Tassion, Vincent},
  title = {Locality of percolation for {Abelian Cayley} graphs},
  journal = {The Annals of Probability},
  volume = {45},
  number = {2},
  pages = {1247--1277},
  year = {2017},
  doi = {10.1214/15-AOP1086},
  eprint = {1312.1946},
  archivePrefix = {arXiv},
  url = {https://doi.org/10.1214/15-AOP1086}
}

@article{ContrerasMartineauTassion2023Locality,
  author = {Contreras, Daniel and Martineau, S{\'e}bastien and Tassion, Vincent},
  title = {Locality of percolation for graphs with polynomial growth},
  journal = {Electronic Communications in Probability},
  volume = {28},
  number = {1},
  pages = {1--9},
  year = {2023},
  doi = {10.1214/22-ECP508},
  eprint = {2205.10253},
  archivePrefix = {arXiv},
  url = {https://doi.org/10.1214/22-ECP508}
}

@inproceedings{OSSS2005,
  author = {O'Donnell, Ryan and Saks, Michael and Schramm, Oded and Servedio, Rocco A.},
  title = {Every decision tree has an influential variable},
  booktitle = {Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science},
  year = {2005},
  pages = {31--39},
  publisher = {IEEE Computer Society},
  doi = {10.1109/SFCS.2005.34}
}

@article{DuminilCopinRaoufiTassion2019,
  author = {Duminil-Copin, Hugo and Raoufi, Aran and Tassion, Vincent},
  title = {Sharp phase transition for the random-cluster and {Potts} models via decision trees},
  journal = {Annals of Mathematics},
  series = {2},
  volume = {189},
  number = {1},
  year = {2019},
  pages = {75--99},
  doi = {10.4007/annals.2019.189.1.2}
}

@article{ChungGrahamFranklShearer1986,
  author = {Chung, F. R. K. and Graham, R. L. and Frankl, P. and Shearer, J. B.},
  title = {Some intersection theorems for ordered sets and graphs},
  journal = {Journal of Combinatorial Theory, Series A},
  volume = {43},
  number = {1},
  year = {1986},
  pages = {23--37},
  doi = {10.1016/0097-3165(86)90019-1}
}

@article{MadimanTetali2010,
  author = {Madiman, Mokshay and Tetali, Prasad},
  title = {Information inequalities for joint distributions, with interpretations and applications},
  journal = {IEEE Transactions on Information Theory},
  volume = {56},
  number = {6},
  year = {2010},
  pages = {2699--2713},
  doi = {10.1109/TIT.2010.2046253}
}

@misc{DLMF,
  author = {{National Institute of Standards and Technology}},
  title = {{NIST Digital Library of Mathematical Functions}},
  year = {2026},
  howpublished = {\url{https://dlmf.nist.gov/}},
  note = {Version 1.2.8, released September 15, 2026; Tables 18.3.1 and 18.6.1; accessed September 24, 2026}
}

@article{BroadbentHammersley1957,
  author={Broadbent, S. R. and Hammersley, J. M.},
  title={Percolation processes. {I}. {Crystals} and mazes},
  journal={Proceedings of the Cambridge Philosophical Society},
  volume={53}, number={3}, pages={629--641}, year={1957},
  doi={10.1017/S0305004100032680}
}

@article{AizenmanBarsky1987,
  author={Aizenman, Michael and Barsky, David J.},
  title={Sharpness of the phase transition in percolation models},
  journal={Communications in Mathematical Physics},
  volume={108}, number={3}, pages={489--526}, year={1987},
  doi={10.1007/BF01212322}
}

@article{Menshikov1986,
  author={Men'shikov, M. V.},
  title={Coincidence of critical points in percolation problems},
  journal={Doklady Akademii Nauk SSSR},
  volume={288}, number={6}, pages={1308--1311}, year={1986},
  note={English translation: Soviet Mathematics Doklady 33 (1986), 856--859},
  url={https://www.mathnet.ru/eng/dan8543}
}

@article{BLPS1999Critical,
  author = {Benjamini, Itai and Lyons, Russell and Peres, Yuval and Schramm, Oded},
  title = {Critical percolation on any nonamenable group has no infinite clusters},
  journal = {The Annals of Probability},
  volume = {27},
  number = {3},
  pages = {1347--1356},
  year = {1999},
  doi = {10.1214/aop/1022677450}
}

@article{Timar2006Nonunimodular,
  author = {Tim{\'a}r, {\'A}d{\'a}m},
  title = {Percolation on nonunimodular transitive graphs},
  journal = {The Annals of Probability},
  volume = {34},
  number = {6},
  pages = {2344--2364},
  year = {2006},
  doi = {10.1214/009117906000000494},
  eprint = {math/0702875},
  archivePrefix = {arXiv},
  primaryClass = {math.PR}
}

@article{DuminilCopinSidoraviciusTassion2016Slabs,
  author={Duminil-Copin, Hugo and Sidoravicius, Vladas and Tassion, Vincent},
  title={Absence of infinite cluster for critical {Bernoulli} percolation on slabs},
  journal={Communications on Pure and Applied Mathematics},
  volume={69}, number={7}, pages={1397--1411}, year={2016},
  doi={10.1002/cpa.21641}, eprint={1401.7130}, archivePrefix={arXiv}
}

@misc{ConnectionInequalities2026,
  author = {{ChatGPT 5.6 Sol} and {Claude Fable 5.1}},
  title = {{Kozma--Nitzan} Connection Inequalities for {Bernoulli} Percolation},
  year = {2026},
  month = sep,
  note = {Prompted by Ahmed Bou-Rabee; public manuscript, updated September 5, 2026},
  url = {https://nitromannitol.github.io/kn1-verification-b80e9/document/kn_summary.pdf}
}


@misc{SiteCriticality2026,
  author = {{ChatGPT 5.6}},
  title = {Percolation at criticality},
  year = {2026},
  month = sep,
  note = {With Claude Opus 5 and GLM 5.3 support; prompted by Ahmed Bou-Rabee. Public exposition and Lean development, September 5, 2026},
  url = {https://nitromannitol.github.io/site-percolation-7d3f2/}
}