## Electronic Journal of Probability

### Convergence of the Critical Finite-Range Contact Process to Super-Brownian Motion Above the Upper Critical Dimension: The Higher-Point Functions

#### Abstract

In this paper, we investigate the contact process higher-point functions which denote the probability that the infection started at the origin at time 0 spreads to an arbitrary number of other individuals at various later times. Together with the results of the two-point function in [16], on which our proofs crucially rely, we prove that the higher-point functions converge to the moment measures of the canonical measure of super-Brownian motion above the upper critical dimension 4. We also prove partial results for in dimension at most 4 in a local mean-field setting. The proof is based on the lace expansion for the time-discretized contact process, which is a version of oriented percolation. For ordinary oriented percolation, we thus reprove the results of [20]. The lace expansion coefficients are shown to obey bounds uniformly in the discretization parameter, which allows us to establish the scaling results also for the contact process We also show that the main term of the vertex factor, which is one of the non-universal constants in the scaling limit, is 1 for oriented percolation, and 2 for the contact process, while the main terms of the other non-universal constants are independent of the discretization parameter. The lace expansion we develop in this paper is adapted to both the higher-point functions and the survival probability. This unified approach makes it easier to relate the expansion coefficients derived in this paper and the expansion coefficients for the survival probability, which will be investigated in a future paper [18].

#### Article information

Source
Electron. J. Probab., Volume 15 (2010), paper no. 27, 801-894.

Dates
Accepted: 11 May 2010
First available in Project Euclid: 1 June 2016

https://projecteuclid.org/euclid.ejp/1464819812

Digital Object Identifier
doi:10.1214/EJP.v15-783

Mathematical Reviews number (MathSciNet)
MR2653947

Zentralblatt MATH identifier
1226.60139

Subjects
Primary: 60j65

Rights

#### Citation

van der Hofstad, Remco; Sakai, Akira. Convergence of the Critical Finite-Range Contact Process to Super-Brownian Motion Above the Upper Critical Dimension: The Higher-Point Functions. Electron. J. Probab. 15 (2010), paper no. 27, 801--894. doi:10.1214/EJP.v15-783. https://projecteuclid.org/euclid.ejp/1464819812

#### References

• Aizenman, Michael; Newman, Charles M. Tree graph inequalities and critical behavior in percolation models. J. Statist. Phys. 36 (1984), no. 1-2, 107–143.
• Bezuidenhout, Carol; Grimmett, Geoffrey. Exponential decay for subcritical contact and percolation processes. Ann. Probab. 19 (1991), no. 3, 984–1009.
• Chen, Lung-Chi; Sakai, Akira. Critical behavior and the limit distribution for long-range oriented percolation. I. Probab. Theory Related Fields 142 (2008), no. 1-2, 151–188.
• Chen, Lung-Chi; Sakai, Akira. Critical behavior and the limit distribution for long-range oriented percolation. I. Probab. Theory Related Fields 142 (2008), no. 1-2, 151–188.
• Durrett, Richard; Perkins, Edwin A. Rescaled contact processes converge to super-Brownian motion in two or more dimensions. Probab. Theory Related Fields 114 (1999), no. 3, 309–399.
• A.M. Etheridge. An Introduction to Superprocesses. American Mathematical Society, Providence, (2000).
• G. Grimmett. Percolation. Springer, Berlin (1999).
• Grimmett, Geoffrey; Hiemer, Philipp. Directed percolation and random walk. In and out of equilibrium (Mambucaba, 2000), 273–297, Progr. Probab., 51, Birkhäuser Boston, Boston, MA, 2002.
• Hara, Takashi; Slade, Gordon. Mean-field critical behaviour for percolation in high dimensions. Comm. Math. Phys. 128 (1990), no. 2, 333–391.
• Hara, Takashi; Slade, Gordon. The scaling limit of the incipient infinite cluster in high-dimensional percolation. I. Critical exponents. J. Statist. Phys. 99 (2000), no. 5-6, 1075–1168. (2001g:82053a)
• Hara, Takashi; Slade, Gordon. The scaling limit of the incipient infinite cluster in high-dimensional percolation. II. Integrated super-Brownian excursion.Probabilistic techniques in equilibrium and nonequilibrium statistical physics. J. Math. Phys. 41 (2000), no. 3, 1244–1293. (2001g:82053b)
• van der Hofstad, Remco. Spread-out oriented percolation and related models above the upper critical dimension: induction and superprocesses. On the nature of isotherms at first order phase transitions for classical lattice models. Spread-out oriented percolation and related models above the upper critical dimension: induction and superprocesses, 91–181 (electronic), Ensaios Mat., 9, Soc. Brasil. Mat., Rio de Janeiro, 2005.
• van der Hofstad, Remco. Infinite canonical super-Brownian motion and scaling limits. Comm. Math. Phys. 265 (2006), no. 3, 547–583.
• van der Hofstad, Remco; den Hollander, Frank; Slade, Gordon. A new inductive approach to the lace expansion for self-avoiding walks. Probab. Theory Related Fields 111 (1998), no. 2, 253–286.
• van der Hofstad, Remco; den Hollander, Frank; Slade, Gordon. Construction of the incipient infinite cluster for spread-out oriented percolation above $4+1$ dimensions. Comm. Math. Phys. 231 (2002), no. 3, 435–461.
• R. van der Hofstad, F. den Hollander, and G. Slade. The survival probability for critical spread-out oriented percolation above $4+1$ dimensions. I. Induction. Probab. Theory Related Fields, 138(2007): 363–389.
• van der Hofstad, Remco; den Hollander, Frank; Slade, Gordon. The survival probability for critical spread-out oriented percolation above $4+1$ dimensions. II. Expansion. Ann. Inst. H. Poincaré Probab. Statist. 43 (2007), no. 5, 509–570.
• van der Hofstad, Remco; Sakai, Akira. Gaussian scaling for the critical spread-out contact process above the upper critical dimension. Electron. J. Probab. 9 (2004), no. 24, 710–769 (electronic).
• R. van der Hofstad and A. Sakai. Critical points for spread-out contact processes and oriented percolation. Probab. Theory Relat. Fields, bf 132 (2005): 438–470.
• R. van der Hofstad and A. Sakai. Convergence of the critical finite-range contact process to super-Brownian motion above the upper critical dimension. II. The survival probability. In preparation.
• van der Hofstad, Remco; Slade, Gordon. A generalised inductive approach to the lace expansion. Probab. Theory Related Fields 122 (2002), no. 3, 389–430.
• van der Hofstad, Remco; Slade, Gordon. Convergence of critical oriented percolation to super-Brownian motion above $4+1$ dimensions. Ann. Inst. H. Poincaré Probab. Statist. 39 (2003), no. 3, 413–485.
• van der Hofstad, Remco; Slade, Gordon. The lace expansion on a tree with application to networks of self-avoiding walks. Adv. in Appl. Math. 30 (2003), no. 3, 471–528.
• M.P. Holmes. Convergence of lattice trees to super-Brownian motion above the critical dimension. Electron. J. Probab.13 (2008): 671–755.
• Holmes, Mark; Perkins, Edwin. Weak convergence of measure-valued processes and $r$-point functions. Ann. Probab. 35 (2007), no. 5, 1769–1782.
• Liggett, Thomas M. Stochastic interacting systems: contact, voter and exclusion processes.Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 324. Springer-Verlag, Berlin, 1999. xii+332 pp. ISBN: 3-540-65995-1
• Sakai, Akira. Mean-field critical behavior for the contact process. J. Statist. Phys. 104 (2001), no. 1-2, 111–143.
• Sakai, Akira. Hyperscaling inequalities for the contact process and oriented percolation. J. Statist. Phys. 106 (2002), no. 1-2, 201–211.
• A. Sakai. Lace expansion for the Ising model. Comm. Math. Phys.272 (2007): 283–344.
• Slade, Gordon. Scaling limits and super-Brownian motion. Notices Amer. Math. Soc. 49 (2002), no. 9, 1056–1067.
• Slade, G. The lace expansion and its applications.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.Lecture Notes in Mathematics, 1879. Springer-Verlag, Berlin, 2006. xiv+228 pp. ISBN: 978-3-540-31189-8; 3-540-31189-0