## Electronic Journal of Probability

### Spatial moments for high-dimensional critical contact process, oriented percolation and lattice trees

#### Abstract

Recently, Holmes and Perkins identified conditions which ensure that for a class of critical lattice models the scaling limit of the range is the range of super-Brownian motion. One of their conditions is an estimate on a spatial moment of order higher than four, which they verified for the sixth moment for spread-out lattice trees in dimensions $d>8$. Chen and Sakai have proved the required moment estimate for spread-out critical oriented percolation in dimensions $d+1>4+1$. We use the lace expansion to prove estimates on all moments for the spread-out critical contact process in dimensions $d>4$, which in particular fulfills the spatial moment condition of Holmes and Perkins. Our method of proof is relatively simple, and, as we show, it applies also to oriented percolation and lattice trees. Via the convergence results of Holmes and Perkins, the upper bounds on the spatial moments can in fact be promoted to asymptotic formulas with explicit constants.

#### Article information

Source
Electron. J. Probab., Volume 24 (2019), paper no. 65, 18 pp.

Dates
Accepted: 26 May 2019
First available in Project Euclid: 22 June 2019

https://projecteuclid.org/euclid.ejp/1561169149

Digital Object Identifier
doi:10.1214/19-EJP327

Mathematical Reviews number (MathSciNet)
MR3978215

Zentralblatt MATH identifier
07089003

#### Citation

Sakai, Akira; Slade, Gordon. Spatial moments for high-dimensional critical contact process, oriented percolation and lattice trees. Electron. J. Probab. 24 (2019), paper no. 65, 18 pp. doi:10.1214/19-EJP327. https://projecteuclid.org/euclid.ejp/1561169149

#### References

• [1] D.J. Barsky and C.C. Wu. Critical exponents for the contact process under the triangle condition. J. Stat. Phys., 91:95–124, (1998).
• [2] C. Bezuidenhout and G. Grimmett. Exponential decay for subcritical contact and percolation processes. Ann. Probab., 19:984–1009, (1991).
• [3] L.-C. Chen and A. Sakai. Critical behavior and the limit distribution for long-range oriented percolation. I. Probab. Theory Related Fields, 142:151–188, (2008).
• [4] L.-C. Chen and A. Sakai. Critical behavior and the limit distribution for long-range oriented percolation. II: Spatial correlation. Probab. Theory Related Fields, 145:435–458, (2009).
• [5] L.-C. Chen and A. Sakai. Asymptotic behavior of the gyration radius for long-range self-avoiding walk and long-range oriented percolation. Ann. Probab., 39:507–548, (2011).
• [6] J.T. Cox, R. Durrett, and E.A. Perkins. Rescaled voter models converge to super-Brownian motion. Ann. Probab., 28:185–234, (2000).
• [7] E. Derbez and G. Slade. The scaling limit of lattice trees in high dimensions. Commun. Math. Phys., 193:69–104, (1998).
• [8] R. Durrett and E.A. Perkins. Rescaled contact processes converge to super-Brownian motion in two or more dimensions. Probab. Theory Related Fields, 114:309–399, (1999).
• [9] P. Flajolet and A. Odlyzko. Singularity analysis of generating functions. SIAM J. Disc. Math., 3:216–240, (1990).
• [10] T. Hara and G. Slade. Mean-field critical behaviour for percolation in high dimensions. Commun. Math. Phys., 128:333–391, (1990).
• [11] T. Hara and G. Slade. On the upper critical dimension of lattice trees and lattice animals. J. Stat. Phys., 59:1469–1510, (1990).
• [12] T. Hara and G. Slade. The number and size of branched polymers in high dimensions. J. Stat. Phys., 67:1009–1038, (1992).
• [13] M. Heydenreich and R. van der Hofstad. Progress in High-Dimensional Percolation and Random Graphs. Springer International Publishing Switzerland, (2017).
• [14] R. van der Hofstad, M. Holmes, and G. Slade. An extension of the inductive approach to the lace expansion. Elect. Comm. Probab., 13:291–301, (2008).
• [15] R. van der Hofstad and A. Sakai. Gaussian scaling for the critical spread-out contact process above the upper critical dimension. Electr. J. Probab., 9:710–769, (2004).
• [16] R. van der Hofstad and A. Sakai. Critical points for spread-out self-avoiding walk, percolation and the contact process. Probab. Theory Related Fields, 132:438–470, (2005).
• [17] R. van der Hofstad and A. Sakai. Convergence of the critical finite-range contact process to super-Brownian motion above the upper critical dimension. Electr. J. Probab., 15:801–894, (2010).
• [18] R. van der Hofstad and G. Slade. A generalised inductive approach to the lace expansion. Probab. Th. Rel. Fields, 122:389–430, (2002).
• [19] R. van der Hofstad and G. Slade. Convergence of critical oriented percolation to super-Brownian motion above $4+1$ dimensions. Ann. Inst. H. Poincaré Probab. Statist., 39:413–485, (2003).
• [20] M. Holmes. Convergence of lattice trees to super-Brownian motion above the critical dimension. Electr. J. Probab., 13:671–755, (2008).
• [21] M. Holmes. Backbone scaling for critical lattice trees in high dimensions. J. Phys. A: Math. Theor., 49:314001, (2016).
• [22] M. Holmes and E. Perkins. On the range of lattice models in high dimensions - extended version. Preprint, https://arxiv.org/abs/1806.08497, (2018).
• [23] M. Holmes and E. Perkins. On the range of lattice models in high dimensions. To appear in Probab. Theory Related Fields.
• [24] B.G. Nguyen and W-S. Yang. Triangle condition for oriented percolation in high dimensions. Ann. Probab., 21:1809–1844, (1993).
• [25] B.G. Nguyen and W-S. Yang. Gaussian limit for critical oriented percolation in high dimensions. J. Stat. Phys., 78:841–876, (1995).
• [26] A. Sakai. Mean-field critical behavior for the contact process. J. Stat. Phys., 104:111–143, (2001).
• [27] G. Slade. Scaling limits and super-Brownian motion. Not. Am. Math. Soc., 49(9):1056–1067, (2002).
• [28] G. Slade. The Lace Expansion and its Applications. Springer, Berlin, (2006). Lecture Notes in Mathematics Vol. 1879. Ecole d’Eté de Probabilités de Saint–Flour XXXIV–2004.
• [29] C.C. Wu. The contact process on a tree: Behavior near the first phase transition. Stoch. Proc. Appl., 57:99–112, (1995).