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

Remco van der Hofstad and Akira Sakai

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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

Permanent link to this document
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

Keywords
contact process mean-field behavior critical exponents super-Brownian motion

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

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


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