{\small \begin{thebibliography}{99} \bibitem{AV08} T.~Antunovi\'c and I.~Veseli\'c. \newblock Sharpness of the phase transition and exponential decay of the subcritical cluster size for percolation on quasi-transitive graphs. \newblock {\em J. Stat. Phys.}, 130:983--1009, 2008. \newblock arXiv:0707.1089. \bibitem{BS96} I.~Benjamini and O.~Schramm. \newblock Percolation beyond $\mathbb{Z}^d$, many questions and a few answers. \newblock {\em Electron. Commun. Probab.}, 1:71--82, 1996. \newblock \href{https://doi.org/10.1214/ECP.v1-978}{doi:10.1214/ECP.v1-978}. \bibitem{BS01} I.~Benjamini and O.~Schramm. \newblock Percolation in the hyperbolic plane. \newblock {\em J. Amer. Math. Soc.}, 14(2):487--507, 2001. \newblock \href{https://doi.org/10.1090/S0894-0347-00-00362-3}{doi:10.1090/S0894-0347-00-00362-3}, arXiv:math/9912233. \bibitem{BK89} R.~M. Burton and M.~Keane. \newblock Density and uniqueness in percolation. \newblock {\em Comm. Math. Phys.}, 121(3):501--505, 1989. \bibitem{Cza13} J.~Czajkowski. \newblock Non-uniqueness phase of Bernoulli percolation on reflection groups for some polyhedra in $\mathbb H^3$. \newblock Preprint, arXiv:1303.5624, 2013. \bibitem{DT16} H.~Duminil-Copin and V.~Tassion. \newblock A new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model. \newblock {\em Comm. Math. Phys.}, 343(2):725--745, 2016. \newblock Correction: {\em Comm. Math. Phys.}, 359(2):821--822, 2018. \bibitem{HPS99} O.~H\"aggstr\"om, Y.~Peres, and R.~H. Schonmann. \newblock Percolation on transitive graphs as a coalescent process: relentless merging followed by simultaneous uniqueness. \newblock In M.~Bramson and R.~Durrett, editors, {\em Perplexing Problems in Probability}, volume~44 of {\em Progr. Probab.}, pages 69--90. Birkh\"auser, Boston, 1999. \newblock \href{https://doi.org/10.1007/978-1-4612-2168-5_4}{doi:10.1007/978-1-4612-2168-5\_4}. \bibitem{Har60} T.~E. Harris. \newblock A lower bound for the critical probability in a certain percolation process. \newblock {\em Proc. Cambridge Philos. Soc.}, 56(1):13--20, 1960. \newblock \href{https://doi.org/10.1017/S0305004100034241}{doi:10.1017/S0305004100034241}. \bibitem{Hut19} T.~Hutchcroft. \newblock Percolation on hyperbolic graphs. \newblock {\em Geom. Funct. Anal.}, 29(3):766--810, 2019. \newblock \href{https://doi.org/10.1007/s00039-019-00498-0}{doi:10.1007/s00039-019-00498-0}, arXiv:1804.10191; numbering as in arXiv:1804.10191v3. \bibitem{HutNU} T.~Hutchcroft. \newblock Non-uniqueness and mean-field criticality for percolation on nonunimodular transitive graphs. \newblock {\em J. Amer. Math. Soc.}, 33(4):1101--1165, 2020. \newblock arXiv:1711.02590; numbering as in arXiv:1711.02590v3. \bibitem{Lal98} S.~P. Lalley. \newblock Percolation on Fuchsian groups. \newblock {\em Ann. Inst. H. Poincar\'e Probab. Statist.}, 34(2):151--177, 1998. \bibitem{Li23} Z.~Li. \newblock Tree embeddings and nonuniqueness in site percolation. \newblock Preprint, arXiv:2304.00923; numbering as in v7, 2026. \bibitem{Li26} Z.~Li. \newblock Planar site percolation, end structure, and the Benjamini--Schramm conjecture. \newblock Preprint, arXiv:2601.09958, 2026. \bibitem{LP16} R.~Lyons and Y.~Peres. \newblock {\em Probability on Trees and Networks}. \newblock Cambridge University Press, 2016. \newblock Numbering and pages as in the online version of 17 January 2026, \url{https://rdlyons.pages.iu.edu/prbtree/book.pdf}. \bibitem{OAINU} OpenAI. \newblock Nonuniqueness of percolation on nonamenable quasi-transitive graphs. \newblock OpenAI Math Release preprint, September 24, 2026. \newblock Available from \href{https://github.com/openai/math/blob/main/preprints/Nonuniqueness-of-percolation-on-nonamenable-quasi-transitive-graphs-September-24-2026/paper.pdf}{\nolinkurl{github.com/openai/math}}. \bibitem{Pei26b} Y.~R. Pei. \newblock Critical site percolation dies on (almost) every quasi-transitive graph. \newblock Preprint, Zenodo, 2026. \newblock \href{https://doi.org/10.5281/zenodo.24023188}{doi:10.5281/zenodo.24023188}. \bibitem{Sal92} M.~Salvatori. \newblock On the norms of group-invariant transition operators on graphs. \newblock {\em J. Theoret. Probab.}, 5(3):563--576, 1992. \bibitem{Sch01} R.~H. Schonmann. \newblock Multiplicity of phase transitions and mean-field criticality on highly non-amenable graphs. \newblock {\em Comm. Math. Phys.}, 219(2):271--322, 2001. \newblock \href{https://doi.org/10.1007/s002200100417}{doi:10.1007/s002200100417}. \bibitem{SW90} P.~M. Soardi and W.~Woess. \newblock Amenability, unimodularity, and the spectral radius of random walks on infinite graphs. \newblock {\em Math. Z.}, 205(3):471--486, 1990. \bibitem{TN26} T.~Tran and T.~M. Nguyen. \newblock Locality of Bernoulli site percolation on transitive graphs. \newblock Preprint, arXiv:2609.07026, 2026; numbering as in v1. \bibitem{vdBK85} J.~van den Berg and H.~Kesten. \newblock Inequalities with applications to percolation and reliability. \newblock {\em J. Appl. Probab.}, 22(3):556--569, 1985. \end{thebibliography} }