Electronic Journal of Probability
- Electron. J. Probab.
- Volume 24 (2019), paper no. 65, 18 pp.
Spatial moments for high-dimensional critical contact process, oriented percolation and lattice trees
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.
Electron. J. Probab., Volume 24 (2019), paper no. 65, 18 pp.
Received: 9 October 2018
Accepted: 26 May 2019
First available in Project Euclid: 22 June 2019
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B27: Critical phenomena 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41]
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