## Electronic Journal of Probability

### Mean-field behavior for nearest-neighbor percolation in $d>10$

#### Abstract

We prove that nearest-neighbor percolation in dimensions $d\geq 11$ displays mean-field behavior by proving that the infrared bound holds, in turn implying the finiteness of the percolation triangle diagram. The finiteness of the triangle implies the existence and mean-field values of various critical exponents, such as $\gamma =1, \beta =1, \delta =2$. We also prove sharp $x$-space asymptotics for the two-point function and the existence of various arm exponents. Such results had previously been obtained in unpublished work by Hara and Slade for nearest-neighbor percolation in dimension $d\geq 19$, so that we bring the dimension above which mean-field behavior is rigorously proved down from $19$ to $11$. Our results also imply sharp bounds on the critical value of nearest-neighbor percolation on $\mathbb{Z} ^d$, which are provably at most $1.31\%$ off in $d=11$. We make use of the general method analyzed in [17], which proposes to use a lace expansion perturbing around non-backtracking random walk. This proof is computer assisted, relying on (1) rigorous numerical upper bounds on various simple random walk integrals as proved by Hara and Slade [25]; and (2) a verification that the numerical conditions in [17] hold true. These two ingredients are implemented in two Mathematica notebooks that can be downloaded from the website of the first author.

The main steps of this paper are (a) to derive a non-backtracking lace expansion for the percolation two-point function; (b) to bound the non-backtracking lace expansion coefficients, thus showing that the general methodology of [17] applies, and (c) to describe the numerical bounds on the coefficients.

#### Article information

Source
Electron. J. Probab., Volume 22 (2017), paper no. 43, 65 pp.

Dates
Accepted: 8 April 2017
First available in Project Euclid: 3 May 2017

https://projecteuclid.org/euclid.ejp/1493777019

Digital Object Identifier
doi:10.1214/17-EJP56

Mathematical Reviews number (MathSciNet)
MR3646069

Zentralblatt MATH identifier
1364.60130

#### Citation

Fitzner, Robert; van der Hofstad, Remco. Mean-field behavior for nearest-neighbor percolation in $d&gt;10$. Electron. J. Probab. 22 (2017), paper no. 43, 65 pp. doi:10.1214/17-EJP56. https://projecteuclid.org/euclid.ejp/1493777019

#### References

• [1] J. Adler, Y. Meir, A. Aharony, and A. Harris. Series study of percolation moments in general dimension. Phys. Rev. B, 41(13):9183–9206, May (1990).
• [2] M. Aizenman and D.J. Barsky. Sharpness of the phase transition in percolation models. Commun. Math. Phys., 108:489–526, (1987).
• [3] M. Aizenman and C.M. Newman. Tree graph inequalities and critical behavior in percolation models. J. Stat. Phys., 36:107–143, (1984).
• [4] D.J. Barsky and M. Aizenman. Percolation critical exponents under the triangle condition. Ann. Probab., 19:1520–1536, (1991).
• [5] R. Bauerschmidt, D. Brydges, and G. Slade. Critical two-point function of the 4-dimensional weakly self-avoiding walk. Comm. Math. Phys., 338(1):169–193, (2015).
• [6] R. Bauerschmidt, D. Brydges, and G Slade. Logarithmic correction for the susceptibility of the 4-dimensional weakly self-avoiding walk: a renormalisation group analysis. Comm. Math. Phys., 337(2):817–877, (2015).
• [7] J. van den Berg and M. Keane. On the continuity of the percolation probability function. In Conference in modern analysis and probability (New Haven, Conn., 1982), volume 26 of Contemp. Math., pages 61–65. Amer. Math. Soc., Providence, RI, (1984).
• [8] J. van den Berg and H. Kesten. Inequalities with applications to percolation and reliability. J. Appl. Prob., 22:556–569, (1985).
• [9] C. Borgs, J. Chayes, R. van der Hofstad, G. Slade, and J. Spencer. Random subgraphs of finite graphs. II. The lace expansion and the triangle condition. Ann. Probab., 33(5):1886–1944, (2005).
• [10] D.C. Brydges and T. Spencer. Self-avoiding walk in 5 or more dimensions. Commun. Math. Phys., 97:125–148, (1985).
• [11] L. Chen, S. Handa, M. Heydenreich, Y. Kamijima, and A. Sakai. An attempt to prove mean-field behavior for nearest-neighbor percolation in 7 dimensions. In preparation.
• [12] H. Duminil-Copin and V. Tassion. A new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model. Commun. Math. Phys., 343(2):725–745, (2016).
• [13] R. Fitzner. www.fitzner.nl/noble/.
• [14] R. Fitzner. Non-backtracking lace expansion. PhD. thesis, TU Eindhoven, (2013).
• [15] R. Fitzner and R. van der Hofstad. Non-backtracking random walk. J. Statist. Phys., 150(2):264–284, (2013).
• [16] R. Fitzner and R. van der Hofstad. Mean-field behavior for nearest-neighbor percolation in $d > 10$: Extended version. arXiv:1506.07977, (2017).
• [17] R. Fitzner and R. van der Hofstad. Generalized approach to the non-backtracking lace expansion. Probab. Theory Related Fields, pages 1–79, to appear (2017).
• [18] P. Grassberger. Critical percolation in high dimensions. Phys. Rev. E, 67:036101, Mar (2003).
• [19] G. Grimmett. Percolation. Springer, Berlin, 2nd edition, (1999).
• [20] S. Handa, Y. Kamijima, and A. Sakai. A survey on the lace expansion for the nearest-neighbor models on the BCC lattice. In preparation.
• [21] T. Hara. Decay of correlations in nearest-neighbor self-avoiding walk, percolation, lattice trees and animals. Ann. Probab., 36(2):530–593, (2008).
• [22] T. Hara and G. Slade. Mean-field critical behaviour for percolation in high dimensions. Commun. Math. Phys., 128:333–391, (1990).
• [23] T. Hara and G. Slade. On the upper critical dimension of lattice trees and lattice animals. J. Stat. Phys., 59:1469–1510, (1990).
• [24] T. Hara and G. Slade. The lace expansion for self-avoiding walk in five or more dimensions. Reviews in Math. Phys., 4:235–327, (1992).
• [25] T. Hara and G. Slade. Self-avoiding walk in five or more dimensions. I. The critical behaviour. Commun. Math. Phys., 147:101–136, (1992).
• [26] T. Hara and G. Slade. Mean-field behaviour and the lace expansion. In G. Grimmett, editor, Probability and Phase Transition, Dordrecht, (1994). Kluwer.
• [27] M. Heydenreich and R. van der Hofstad. Progress in high-dimensional percolation and random graphs. Lecture notes for the CRM-PIMS Summer School in Probability, Preprint (2016).
• [28] M. Heydenreich, R. van der Hofstad, and T. Hulshof. High-dimensional incipient infinite clusters revisited. J. Stat. Phys., 155(5):966–1025, (2014).
• [29] M. Heydenreich, R. van der Hofstad, and A. Sakai. Mean-field behavior for long- and finite range Ising model, percolation and self-avoiding walk. J. Statist. Phys., 132(5):1001–1049, (2008).
• [30] R. van der Hofstad, F. den Hollander, and G. Slade. A new inductive approach to the lace expansion for self-avoiding walks. Probab. Theory Related Fields, 111(2):253–286, (1998).
• [31] R. van der Hofstad, F. den Hollander, and G. Slade. The survival probability for critical spread-out oriented percolation above 4+1 dimensions. II. Expansion. Ann. Inst. H. Poincaré Probab. Statist., 5(5):509–570, (2007).
• [32] R. van der Hofstad and A.A. Járai. The incipient infinite cluster for high-dimensional unoriented percolation. J. Statist. Phys., 114(3–4):625–663, (2004).
• [33] R. van der Hofstad and A. Sakai. Gaussian scaling for the critical spread-out contact process above the upper critical dimension. Electron. J. Probab., 9:710–769 (electronic), (2004).
• [34] R. van der Hofstad and G. Slade. A generalised inductive approach to the lace expansion. Probab. Theory Related Fields, 122(3):389–430, (2002).
• [35] B. D. Hughes. Random walks and random environments. Vol. 2. Oxford Science Publications. The Clarendon Press Oxford University Press, New York, (1996).
• [36] A.A. Járai. Incipient infinite percolation clusters in 2D. Ann. Probab., 31(1):444–485, (2003).
• [37] A.A. Járai. Invasion percolation and the incipient infinite cluster in 2D. Comm. Math. Phys., 236(2):311–334, (2003).
• [38] H. Kesten. The incipient infinite cluster in two-dimensional percolation. Probab. Theory Related Fields, 73(3):369–394, (1986).
• [39] G. Kozma and A. Nachmias. The Alexander-Orbach conjecture holds in high dimensions. Inventiones Mathematicae, 178(3):635–654, (2009).
• [40] G. Kozma and A. Nachmias. Arm exponents in high dimensional percolation. J. Amer. Math. Soc., 24(2):375–409, (2011).
• [41] N. Madras and G. Slade. The Self-Avoiding Walk. Birkhäuser, Boston, (1993).
• [42] M.V. Menshikov. Coincidence of critical points in percolation problems. Soviet Mathematics, Doklady, 33:856–859, (1986).
• [43] B.G. Nguyen and W-S. Yang. Triangle condition for oriented percolation in high dimensions. Ann. Probab., 21:1809–1844, (1993).
• [44] B.G. Nguyen and W-S. Yang. Gaussian limit for critical oriented percolation in high dimensions. J. Stat. Phys., 78(3):841–876, (1995).
• [45] L. Russo. A note on percolation. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 43(1):39–48, (1978).
• [46] L. Russo. On the critical percolation probabilities. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 56(2):229–237, (1981).
• [47] A. Sakai. Mean-field critical behavior for the contact process. J. Statist. Phys., 104(1–2):111–143, (2001).
• [48] A. Sakai. Lace expansion for the Ising model. Comm. Math. Phys., 272(2):283–344, (2007).
• [49] G. Slade. The diffusion of self-avoiding random walk in high dimensions. Commun. Math. Phys., 110:661–683, (1987).
• [50] G. Slade. The lace expansion and its applications, volume 1879 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, (2006).