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Locality of the critical probability for transitive graphs of exponential growth. (English) Zbl 07226363
Summary: Around 2008, Schramm conjectured that the critical probabilities for Bernoulli bond percolation satisfy the following continuity property: If \((G_n)_{n\geq 1}\) is a sequence of transitive graphs converging locally to a transitive graph \(G\) and \(\limsup_{n\to \infty}p_c(G_n)<1\), then \(p_c(G_n)\to p_c(G)\) as \(n\to\infty\). We verify this conjecture under the additional hypothesis that there is a uniform exponential lower bound on the volume growth of the graphs in question. The result is new even in the case that the sequence of graphs is uniformly nonamenable.
In the unimodular case, this result is obtained as a corollary to the following theorem of independent interest: For every \(g > 1\) and \(M< \infty\), there exist positive constants \(C=C(g,M)\) and \(\delta=\delta (g,M)\) such that if \(G\) is a transitive unimodular graph with degree at most \(M\) and growth \(\text{gr}(G):=\inf_{r\geq 1}|B(o,r)|^{1/r}\geq g\), then \[ \mathbf{P}_{p_c}\bigl(\vert K_o\vert\geq n\bigr)\leq Cn^{-\delta} \] for every \(n\geq 1\), where \(K_o\) is the cluster of the root vertex \(o\). The proof of this inequality makes use of new universal bounds on the probabilities of certain two-arm events, which hold for every unimodular transitive graph.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation
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References:
[1] Aizenman, M. and Barsky, D. J. (1987). Sharpness of the phase transition in percolation models. Comm. Math. Phys. 108 489-526. · Zbl 0618.60098
[2] Aizenman, M., Kesten, H. and Newman, C. M. (1987). Uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation. Comm. Math. Phys. 111 505-531. · Zbl 0642.60102
[3] Aldous, D. and Lyons, R. (2007). Processes on unimodular random networks. Electron. J. Probab. 12 1454-1508. · Zbl 1131.60003
[4] Benjamini, I. (2013). Euclidean vs. graph metric. In Erdös Centennial. Bolyai Soc. Math. Stud. 25 35-57. János Bolyai Math. Soc., Budapest. · Zbl 1293.05199
[5] Benjamini, I. and Curien, N. (2012). Ergodic theory on stationary random graphs. Electron. J. Probab. 17 no. 93, 20. · Zbl 1278.05222
[6] Benjamini, I., Lyons, R., Peres, Y. and Schramm, O. (1999). Critical percolation on any nonamenable group has no infinite clusters. Ann. Probab. 27 1347-1356. · Zbl 0961.60015
[7] Benjamini, I., Nachmias, A. and Peres, Y. (2011). Is the critical percolation probability local? Probab. Theory Related Fields 149 261-269. · Zbl 1230.60099
[8] Benjamini, I. and Schramm, O. (2011). Percolation beyond \(\mathbb{Z}^d \), many questions and a few answers [MR1423907]. In Selected Works of Oded Schramm. Volume 1, 2. Sel. Works Probab. Stat. 679-690. Springer, New York. · Zbl 0890.60091
[9] Burton, R. M. and Keane, M. (1989). Density and uniqueness in percolation. Comm. Math. Phys. 121 501-505. · Zbl 0662.60113
[10] Cerf, R. (2015). A lower bound on the two-arms exponent for critical percolation on the lattice. Ann. Probab. 43 2458-2480. · Zbl 1356.60163
[11] Chayes, J. T. and Chayes, L. (1986). Inequality for the infinite-cluster density in Bernoulli percolation. Phys. Rev. Lett. 56 1619-1622.
[12] Curien, N. (2017). Random graphs: The local convergence point of view. Unpublished lecture notes. Available at: https://www.math.u-psud.fr/ curien/cours/cours-RG-V3.pdf.
[13] Duminil-Copin, H., Goswami, S., Raoufi, A., Severo, F. and Yadin, A. (2018). Existence of phase transition for percolation using the Gaussian free field. Available at arXiv:1806.07733.
[14] Duminil-Copin, H. and Tassion, V. (2016). A new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model. Comm. Math. Phys. 343 725-745. · Zbl 1342.82026
[15] Duminil-Copin, H. and Tassion, V. (2017). A note on Schramm’s locality conjecture for random-cluster models. Available at arXiv:1707.07626. · Zbl 1357.82011
[16] Gandolfi, A., Grimmett, G. and Russo, L. (1988). On the uniqueness of the infinite cluster in the percolation model. Comm. Math. Phys. 114 549-552. · Zbl 0649.60104
[17] Grimmett, G. (1999). Percolation, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 321. Springer, Berlin. · Zbl 0926.60004
[18] Grimmett, G. R. and Li, Z. (2014). Locality of connective constants. · Zbl 1397.05076
[19] Grimmett, G. R. and Li, Z. (2017). Self-avoiding walks and amenability. Electron. J. Combin. 24 Paper 4.38, 24. · Zbl 1376.05068
[20] Grimmett, G. R. and Marstrand, J. M. (1990). The supercritical phase of percolation is well behaved. Proc. Roy. Soc. London Ser. A 430 439-457. · Zbl 0711.60100
[21] Heydenreich, M. and van der Hofstad, R. (2017). Progress in High-Dimensional Percolation and Random Graphs. CRM Short Courses. Springer, Cham; Centre de Recherches Mathématiques, Montreal, QC. · Zbl 1445.60003
[22] Hutchcroft, T. (2016). Critical percolation on any quasi-transitive graph of exponential growth has no infinite clusters. C. R. Math. Acad. Sci. Paris 354 944-947. · Zbl 1351.60128
[23] Hutchcroft, T. (2017). Non-uniqueness and mean-field criticality for percolation on nonunimodular transitive graphs. Available at arXiv:1711.02590.
[24] Hutchcroft, T. (2019). Percolation on hyperbolic graphs. Geom. Funct. Anal. 29 766-810. · Zbl 07073753
[25] Hutchcroft, T. (2019). Self-avoiding walk on nonunimodular transitive graphs. Ann. Probab. 47 2801-2829. · Zbl 1448.60187
[26] Lyons, R. and Peres, Y. (2016). Probability on Trees and Networks. Cambridge Series in Statistical and Probabilistic Mathematics 42. Cambridge Univ. Press, New York.
[27] Martineau, S. and Tassion, V. (2017). Locality of percolation for Abelian Cayley graphs. Ann. Probab. 45 1247-1277. · Zbl 1388.60165
[28] Mohar, B. and Woess, W. (1989). A survey on spectra of infinite graphs. Bull. Lond. Math. Soc. 21 209-234. · Zbl 0645.05048
[29] Nachmias, A. and Peres, Y. (2012). Non-amenable Cayley graphs of high girth have \(p_c<p_u\) and mean-field exponents. Electron. Commun. Probab. 17 no. 57, 8. · Zbl 1302.82056
[30] Pak, I. and Smirnova-Nagnibeda, T. (2000). On non-uniqueness of percolation on nonamenable Cayley graphs. C. R. Acad. Sci. Paris Sér. I Math. 330 495-500. · Zbl 0947.43003
[31] Pete, G. (2014). Probability and geometry on groups. Available at http://www.math.bme.hu/ gabor/PGG.pdf.
[32] Schonmann, R. H. (2001). Multiplicity of phase transitions and mean-field criticality on highly non-amenable graphs. Comm. Math. Phys. 219 271-322. · Zbl 1038.82037
[33] Soardi, P. M. and Woess, W. (1990). Amenability, unimodularity, and the spectral radius of random walks on infinite graphs. Math. Z. 205 471-486. · Zbl 0693.43001
[34] Song, H., Xiang, K.-N. and Zhu, S.-C.-H. (2014). Locality of percolation critical probabilities: Uniformly nonamenable case. Preprint. Available at arXiv:1410.2453.
[35] Timár, Á. (2006). Percolation on nonunimodular transitive graphs. Ann. Probab. 34 2344-2364. · Zbl 1114.60083
[36] Werner, W. · Zbl 1180.82003
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