Abstract
The contact process is a model of spread of an infectious disease. Combining with the result of ref. 1, we prove that the critical exponents take on the mean-field values for sufficiently high dimensional nearest-neighbor models and for sufficiently spread-out models with d>4:θ(λ)≈λ−λ c as λ↓λ c and χ(λ)≈(λ c−λ)−1 as λ↑λ c, where θ(λ) and χ(λ) are the spread probability and the susceptibility of the infection respectively, and λ c is the critical infection rate. Our results imply that the upper critical dimension for the contact process is at most 4.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
D. J. Barsky and C. C. Wu, Critical exponents for the contact process under the triangle condition, J. Stat. Phys. 91:95–124 (1998).
C. Bezuidenhout and G. Grimmett, The critical contact process dies out, Ann. Probab. 18:1462–1482 (1990).
C. Bezuidenhout and G. Grimmett, Exponential decay for subcritical contact and percolation processes, Ann. Probab. 19:984–1009 (1991).
M. Bramson, R. Durrett, and G. Swindle, Statistical mechanics of crabgrass, Ann. Probab. 17:444–481 (1989).
G. Grimmett, Percolation (Springer-Verlag, 1999).
T. Hara, Mean-field critical behaviour for correlation length for percolation in high dimensions, Probab. Theory Rel. Fields 86:337–385 (1990).
T. Hara and G. Slade, Mean-field critical behaviour for percolation in high dimensions, Commun. Math. Phys. 128:333–391 (1990).
T. E. Harris, Contact interactions on a lattice, Ann. Probab. 2:969–988 (1974).
N. Madras and G. Slade, The Self-Avoiding Walk (Birkhäuser, 1993).
B. G. Nguyen and W.-S. Yang, Triangle condition for oriented percolation in high dimensions, Ann. Probab. 21:1809–1844 (1993).
B. G. Nguyen and W.-S. Yang, Gaussian limit for critical oriented percolation in high dimensions, J. Stat. Phys. 78:841–876 (1995).
R. H. Schonmann, The triangle condition for contact processes on homogeneous trees, J. Stat. Phys. 90:1429–1440 (1998).
H. Tasaki, Geometric critical exponent inequalities for general random cluster models, J. Stat. Phys. 49:841–847 (1987).
C. C. Wu, The contact process on a tree: Behavior near the first phase transition, Stoch. Proc. Appl. 57:99–112 (1995).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Sakai, A. Mean-Field Critical Behavior for the Contact Process. Journal of Statistical Physics 104, 111–143 (2001). https://doi.org/10.1023/A:1010320523031
Issue Date:
DOI: https://doi.org/10.1023/A:1010320523031