## Nonuniqueness and mean-field criticality for percolation on nonunimodular transitive graphs

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Tom Hutchcroft
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## Abstract:

We study Bernoulli bond percolation on nonunimodular quasi-transitive graphs, and more generally graphs whose automorphism group has a nonunimodular quasi-transitive subgroup. We prove that percolation on any such graph has a nonempty phase in which there are infinite *light* clusters, which implies the existence of a nonempty phase in which there are *infinitely many* infinite clusters. That is, we show that $p_c<p_h \leq p_u$ for any such graph. This answers a question of Häggström, Peres, and Schonmann (1999), and verifies the nonunimodular case of a well-known conjecture of Benjamini and Schramm (1996). We also prove that the triangle condition holds at criticality on any such graph, which implies that various critical exponents exist and take their mean-field values.

All our results apply, for example, to the product $T_k\times \mathbb {Z}^d$ of a $k$-regular tree with $\mathbb {Z}^d$ for $k\geq 3$ and $d \geq 1$, for which these results were previously known only for large $k$. Furthermore, our methods also enable us to establish the basic topological features of the phase diagram for *anisotropic* percolation on such products, in which tree edges and $\mathbb {Z}^d$ edges are given different retention probabilities. These features had only previously been established for $d=1$, $k$ large.

## References

- Michael Aizenman and David J. Barsky,
*Sharpness of the phase transition in percolation models*, Comm. Math. Phys.**108**(1987), no. 3, 489–526. MR**874906**, DOI 10.1007/BF01212322 - M. Aizenman, H. Kesten, and C. M. Newman,
*Uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation*, Comm. Math. Phys.**111**(1987), no. 4, 505–531. MR**901151**, DOI 10.1007/BF01219071 - Michael Aizenman and Charles M. Newman,
*Tree graph inequalities and critical behavior in percolation models*, J. Statist. Phys.**36**(1984), no. 1-2, 107–143. MR**762034**, DOI 10.1007/BF01015729 - Omer Angel and Tom Hutchcroft,
*Counterexamples for percolation on unimodular random graphs*, Unimodularity in randomly generated graphs, Contemp. Math., vol. 719, Amer. Math. Soc., [Providence], RI, [2018] ©2018, pp. 11–28. MR**3880008**, DOI 10.1090/conm/719/14465 - Tonći Antunović and Ivan Veselić,
*Sharpness of the phase transition and exponential decay of the subcritical cluster size for percolation and quasi-transitive graphs*, J. Stat. Phys.**130**(2008), no. 5, 983–1009. MR**2384072**, DOI 10.1007/s10955-007-9459-x - Richard Arratia, Skip Garibaldi, and Alfred W. Hales,
*The van den Berg–Kesten-Reimer operator and inequality for infinite spaces*, Bernoulli**24**(2018), no. 1, 433–448. MR**3706764**, DOI 10.3150/16-BEJ883 - Eric Babson and Itai Benjamini,
*Cut sets and normed cohomology with applications to percolation*, Proc. Amer. Math. Soc.**127**(1999), no. 2, 589–597. MR**1622785**, DOI 10.1090/S0002-9939-99-04995-3 - D. J. Barsky and M. Aizenman,
*Percolation critical exponents under the triangle condition*, Ann. Probab.**19**(1991), no. 4, 1520–1536. MR**1127713**, DOI 10.1214/aop/1176990221 - Itai Benjamini and Nicolas Curien,
*Ergodic theory on stationary random graphs*, Electron. J. Probab.**17**(2012), no. 93, 20. MR**2994841**, DOI 10.1214/EJP.v17-2401 - Itai Benjamini, Russell Lyons, Yuval Peres, and Oded Schramm,
*Critical percolation on any nonamenable group has no infinite clusters*, Ann. Probab.**27**(1999), no. 3, 1347–1356. MR**1733151**, DOI 10.1214/aop/1022677450 - Itai Benjamini and Oded Schramm,
*Percolation beyond $\mathbf Z^d$, many questions and a few answers*, Electron. Comm. Probab.**1**(1996), no. 8, 71–82. MR**1423907**, DOI 10.1214/ECP.v1-978 - Itai Benjamini and Oded Schramm,
*Percolation in the hyperbolic plane*, J. Amer. Math. Soc.**14**(2001), no. 2, 487–507. MR**1815220**, DOI 10.1090/S0894-0347-00-00362-3 - R. M. Burton and M. Keane,
*Density and uniqueness in percolation*, Comm. Math. Phys.**121**(1989), no. 3, 501–505. MR**990777**, DOI 10.1007/BF01217735 - M. de la Salle and R. Tessera,
*Characterizing a vertex-transitive graph by a large ball*, Journal of Topology, 12(3):704–742, 2019. - Reinhard Diestel and Imre Leader,
*A conjecture concerning a limit of non-Cayley graphs*, J. Algebraic Combin.**14**(2001), no. 1, 17–25. MR**1856226**, DOI 10.1023/A:1011257718029 - H. Duminil-Copin,
*Lectures on the Ising and Potts models on the hypercubic lattice*, 2017. - Hugo Duminil-Copin and Vincent Tassion,
*A new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model*, Comm. Math. Phys.**343**(2016), no. 2, 725–745. MR**3477351**, DOI 10.1007/s00220-015-2480-z - R. Durrett and B. Nguyen,
*Thermodynamic inequalities for percolation*, Comm. Math. Phys.**99**(1985), no. 2, 253–269. MR**790737**, DOI 10.1007/BF01212282 - Robert Fitzner and Remco van der Hofstad,
*Mean-field behavior for nearest-neighbor percolation in $d>10$*, Electron. J. Probab.**22**(2017), Paper No. 43, 65. MR**3646069**, DOI 10.1214/17-EJP56 - D. Gaboriau,
*Invariant percolation and harmonic Dirichlet functions*, Geom. Funct. Anal.**15**(2005), no. 5, 1004–1051. MR**2221157**, DOI 10.1007/s00039-005-0539-2 - A. Gandolfi, M. S. Keane, and C. M. Newman,
*Uniqueness of the infinite component in a random graph with applications to percolation and spin glasses*, Probab. Theory Related Fields**92**(1992), no. 4, 511–527. MR**1169017**, DOI 10.1007/BF01274266 - Geoffrey Grimmett,
*Percolation*, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 321, Springer-Verlag, Berlin, 1999. MR**1707339**, DOI 10.1007/978-3-662-03981-6 - G. R. Grimmett and C. M. Newman,
*Percolation in $\infty +1$ dimensions*, Disorder in physical systems, Oxford Sci. Publ., Oxford Univ. Press, New York, 1990, pp. 167–190. MR**1064560** - Olle Häggström,
*Percolation beyond $\Bbb Z^d$: the contributions of Oded Schramm*, Ann. Probab.**39**(2011), no. 5, 1668–1701. MR**2884871**, DOI 10.1214/10-AOP563 - Olle Häggström and Johan Jonasson,
*Uniqueness and non-uniqueness in percolation theory*, Probab. Surv.**3**(2006), 289–344. MR**2280297**, DOI 10.1214/154957806000000096 - Olle Häggström and Yuval Peres,
*Monotonicity of uniqueness for percolation on Cayley graphs: all infinite clusters are born simultaneously*, Probab. Theory Related Fields**113**(1999), no. 2, 273–285. MR**1676835**, DOI 10.1007/s004400050208 - Olle Häggström, Yuval Peres, and Roberto H. Schonmann,
*Percolation on transitive graphs as a coalescent process: relentless merging followed by simultaneous uniqueness*, Perplexing problems in probability, Progr. Probab., vol. 44, Birkhäuser Boston, Boston, MA, 1999, pp. 69–90. MR**1703125** - Takashi Hara,
*Decay of correlations in nearest-neighbor self-avoiding walk, percolation, lattice trees and animals*, Ann. Probab.**36**(2008), no. 2, 530–593. MR**2393990**, DOI 10.1214/009117907000000231 - Takashi Hara and Gordon Slade,
*Mean-field critical behaviour for percolation in high dimensions*, Comm. Math. Phys.**128**(1990), no. 2, 333–391. MR**1043524**, DOI 10.1007/BF02108785 - Takashi Hara and Gordon Slade,
*Mean-field behaviour and the lace expansion*, Probability and phase transition (Cambridge, 1993) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 420, Kluwer Acad. Publ., Dordrecht, 1994, pp. 87–122. MR**1283177** - Takashi Hara, Remco van der Hofstad, and Gordon Slade,
*Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models*, Ann. Probab.**31**(2003), no. 1, 349–408. MR**1959796**, DOI 10.1214/aop/1046294314 - T. E. Harris,
*A lower bound for the critical probability in a certain percolation process*, Proc. Cambridge Philos. Soc.**56**(1960), 13–20. MR**115221**, DOI 10.1017/S0305004100034241 - J. Hermon and T. Hutchcroft,
*Supercritical percolation on nonamenable graphs: Isoperimetry, analyticity, and exponential decay of the cluster size distribution*, 2019. Preprint. Available at arXiv:1808.08940. - Markus Heydenreich and Remco van der Hofstad,
*Progress in high-dimensional percolation and random graphs*, CRM Short Courses, Springer, Cham; Centre de Recherches Mathématiques, Montreal, QC, 2017. MR**3729454**, DOI 10.1007/978-3-319-62473-0 - Tom Hutchcroft,
*Critical percolation on any quasi-transitive graph of exponential growth has no infinite clusters*, C. R. Math. Acad. Sci. Paris**354**(2016), no. 9, 944–947. MR**3535351**, DOI 10.1016/j.crma.2016.07.013 - T. Hutchcroft,
*Locality of the critical probability for transitive graphs of exponential growth*, Annals of Probability, 2018. To appear. Available at arXiv:1808.08940. - T. Hutchcroft,
*The ${L}^2$ boundedness condition in nonamenable percolation*, 2019. Preprint. Available at arXiv:1901.10363. - Tom Hutchcroft,
*Percolation on hyperbolic graphs*, Geom. Funct. Anal.**29**(2019), no. 3, 766–810. MR**3962879**, DOI 10.1007/s00039-019-00498-0 - Tom Hutchcroft,
*Self-avoiding walk on nonunimodular transitive graphs*, Ann. Probab.**47**(2019), no. 5, 2801–2829. MR**4021237**, DOI 10.1214/18-AOP1322 - Tom Hutchcroft,
*Statistical physics on a product of trees*, Ann. Inst. Henri Poincaré Probab. Stat.**55**(2019), no. 2, 1001–1010. MR**3949961**, DOI 10.1214/18-aihp906 - Gady Kozma,
*Percolation on a product of two trees*, Ann. Probab.**39**(2011), no. 5, 1864–1895. MR**2884876**, DOI 10.1214/10-AOP618 - Gady Kozma,
*The triangle and the open triangle*, Ann. Inst. Henri Poincaré Probab. Stat.**47**(2011), no. 1, 75–79 (English, with English and French summaries). MR**2779397**, DOI 10.1214/09-AIHP352 - Gady Kozma and Asaf Nachmias,
*The Alexander-Orbach conjecture holds in high dimensions*, Invent. Math.**178**(2009), no. 3, 635–654. MR**2551766**, DOI 10.1007/s00222-009-0208-4 - Gady Kozma and Asaf Nachmias,
*Arm exponents in high dimensional percolation*, J. Amer. Math. Soc.**24**(2011), no. 2, 375–409. MR**2748397**, DOI 10.1090/S0894-0347-2010-00684-4 - Steven P. Lalley,
*Percolation on Fuchsian groups*, Ann. Inst. H. Poincaré Probab. Statist.**34**(1998), no. 2, 151–177 (English, with English and French summaries). MR**1614583**, DOI 10.1016/S0246-0203(98)80022-8 - Russell Lyons,
*Phase transitions on nonamenable graphs*, J. Math. Phys.**41**(2000), no. 3, 1099–1126. Probabilistic techniques in equilibrium and nonequilibrium statistical physics. MR**1757952**, DOI 10.1063/1.533179 - Russell Lyons,
*Fixed price of groups and percolation*, Ergodic Theory Dynam. Systems**33**(2013), no. 1, 183–185. MR**3009109**, DOI 10.1017/S0143385711000927 - Russell Lyons and Yuval Peres,
*Probability on trees and networks*, Cambridge Series in Statistical and Probabilistic Mathematics, vol. 42, Cambridge University Press, New York, 2016. MR**3616205**, DOI 10.1017/9781316672815 - Russell Lyons, Yuval Peres, and Oded Schramm,
*Minimal spanning forests*, Ann. Probab.**34**(2006), no. 5, 1665–1692. MR**2271476**, DOI 10.1214/009117906000000269 - Russell Lyons and Oded Schramm,
*Indistinguishability of percolation clusters*, Ann. Probab.**27**(1999), no. 4, 1809–1836. MR**1742889**, DOI 10.1214/aop/1022677549 - Asaf Nachmias and Yuval Peres,
*Non-amenable Cayley graphs of high girth have $p_c<p_u$ and mean-field exponents*, Electron. Commun. Probab.**17**(2012), no. 57, 8. MR**3005730**, DOI 10.1214/ECP.v17-2139 - Charles M. Newman,
*Another critical exponent inequality for percolation: $\beta \geq 2/\delta$*, Proceedings of the symposium on statistical mechanics of phase transitions—mathematical and physical aspects (Trebon, 1986), 1987, pp. 695–699. MR**912497**, DOI 10.1007/BF01206153 - Bao Gia Nguyen,
*Gap exponents for percolation processes with triangle condition*, J. Statist. Phys.**49**(1987), no. 1-2, 235–243. MR**923855**, DOI 10.1007/BF01009960 - Igor Pak and Tatiana Smirnova-Nagnibeda,
*On non-uniqueness of percolation on nonamenable Cayley graphs*, C. R. Acad. Sci. Paris Sér. I Math.**330**(2000), no. 6, 495–500 (English, with English and French summaries). MR**1756965**, DOI 10.1016/S0764-4442(00)00211-1 - R. Paley and A. Zygmund. A note on analytic functions in the unit circle. In
*Mathematical Proceedings of the Cambridge Philosophical Society*, volume 28, pages 266–272. Cambridge Univ Press, 1932. - Yuval Peres,
*Percolation on nonamenable products at the uniqueness threshold*, Ann. Inst. H. Poincaré Probab. Statist.**36**(2000), no. 3, 395–406 (English, with English and French summaries). MR**1770624**, DOI 10.1016/S0246-0203(00)00130-8 - Yuval Peres, Gábor Pete, and Ariel Scolnicov,
*Critical percolation on certain nonunimodular graphs*, New York J. Math.**12**(2006), 1–18. MR**2217160** - David Reimer,
*Proof of the van den Berg-Kesten conjecture*, Combin. Probab. Comput.**9**(2000), no. 1, 27–32. MR**1751301**, DOI 10.1017/S0963548399004113 - Artëm Sapozhnikov,
*Upper bound on the expected size of the intrinsic ball*, Electron. Commun. Probab.**15**(2010), 297–298. MR**2670196**, DOI 10.1214/ECP.v15-1553 - Roberto H. Schonmann,
*Stability of infinite clusters in supercritical percolation*, Probab. Theory Related Fields**113**(1999), no. 2, 287–300. MR**1676831**, DOI 10.1007/s004400050209 - Roberto H. Schonmann,
*Multiplicity of phase transitions and mean-field criticality on highly non-amenable graphs*, Comm. Math. Phys.**219**(2001), no. 2, 271–322. MR**1833805**, DOI 10.1007/s002200100417 - Roberto H. Schonmann,
*Mean-field criticality for percolation on planar non-amenable graphs*, Comm. Math. Phys.**225**(2002), no. 3, 453–463. MR**1888869**, DOI 10.1007/s002200100587 - G. Slade,
*The lace expansion and its applications*, Lecture Notes in Mathematics, vol. 1879, Springer-Verlag, Berlin, 2006. Lectures from the 34th Summer School on Probability Theory held in Saint-Flour, July 6–24, 2004; Edited and with a foreword by Jean Picard. MR**2239599** - Pengfei Tang,
*Heavy Bernoulli-percolation clusters are indistinguishable*, Ann. Probab.**47**(2019), no. 6, 4077–4115. MR**4038049**, DOI 10.1214/19-aop1354 - Ádám Timár,
*Percolation on nonunimodular transitive graphs*, Ann. Probab.**34**(2006), no. 6, 2344–2364. MR**2294985**, DOI 10.1214/009117906000000494 - Ádám Timár,
*Cutsets in infinite graphs*, Combin. Probab. Comput.**16**(2007), no. 1, 159–166. MR**2286517**, DOI 10.1017/S0963548306007838 - V. I. Trofimov,
*Groups of automorphisms of graphs as topological groups*, Mat. Zametki**38**(1985), no. 3, 378–385, 476 (Russian). MR**811571** - J. van den Berg and U. Fiebig,
*On a combinatorial conjecture concerning disjoint occurrences of events*, Ann. Probab.**15**(1987), no. 1, 354–374. MR**877608** - K. Yamamoto,
*An upper bound for the critical probability on the cartesian product graph of a regular tree and a line*, Kobe Journal of Mathematics, pages 43–56, 2019.

## Additional Information

**Tom Hutchcroft**- Affiliation: Statslab, Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambride, United Kingdom
- MR Author ID: 1126713
- Received by editor(s): November 29, 2017
- Received by editor(s) in revised form: July 9, 2019, and February 23, 2020
- Published electronically: September 23, 2020
- Additional Notes: This work mainly took place while the author was a Ph.D. student at the University of British Columbia, during which time he was supported by a Microsoft Research Ph.D. Fellowship.
- © Copyright 2020 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**33**(2020), 1101-1165 - MSC (2010): Primary 60B99, 60K35
- DOI: https://doi.org/10.1090/jams/953
- MathSciNet review: 4155221