We consider the critical spread-out contact process in Z^d with d≥1, whose infection range is denoted by L≥1.| The two-point function τ_t(x) is the probability that x in Z^d is infected at time t by the infected individual located at the origin o in Z^d at time 0. We prove Gaussian behaviour for the two-point function with L≥L₀ for some finite L₀= L₀(d) for d>4.| When d≤4, we also perform a local mean-field limit to obtain Gaussian behaviour for τ_tT(x) with t>0 fixed and T tending to infinity when the infection range depends on T in such a way that L_T=LT^b for any b>(4-d)/2d.

The proof is based on the lace expansion and an adaptation of the inductive approach applied to the discretized contact process.| We prove the existence of several critical exponents and show that they take on their respective mean-field values.| The results in this paper provide crucial ingredients to prove convergence of the finite-dimensional distributions for the contact process towards those for the canonical measure of super-Brownian motion, which we defer to a sequel of this paper.

The results in this paper also apply to oriented percolation, for which we reprove some of the results in [20] and extend the results to the local mean-field setting described above when d≤4.

[1] M. Aizenman and C. M. Newman. Tree graph inequalities and critical behavior in percolation models.  J. Statist. Phys. 36 (1984): 107--143.

 

[2] D. J. Barsky and M. Aizenman. Percolation critical exponents under the triangle condition. Ann. Probab. 19 (1991): 1520--1536.

 

[3] D. J. Barsky and C. C. Wu. Critical exponents for the contact process under the triangle condition. J. Statist. Phys. 91 (1998): 95--124.

 

[4] C. Bezuidenhout and G. Grimmett. Exponential decay for subcritical contact and percolation processes. Ann. Probab. 19 (1991): 984--1009.

 

[5] E. Bolthausen and C. Ritzmann. Strong pointwise estimates for the weakly self-avoiding walk. To appear in Ann. Probab.

 

[6] C. Borgs, J. T. Chayes and D. Randall. The van den Berg-Kesten-Reimer inequality. Perplexing Problems in Probability: Festschrift in honor of Harry Kesten (eds., M. Bramson and R. Durrett). Birkhäuser (1999): 159-173.

 

[7] D.C. Brydges and T. Spencer. Self-avoiding walk in 5 or more dimensions. Commun. Math. Phys., 97 (1985):125--148.

 

[8] R. Durrett and E. Perkins. Rescaled contact processes converge to super-Brownian motion in two or more dimensions. Probab. Th. Rel. Fields 114 (1999): 309--399.

 

[9] G. Grimmett. Percolation.  Springer, Berlin (1999).

 

[10] G. Grimmett and P. Hiemer. Directed percolation and random walk. In and Out of Equilibrium (ed., V. Sidoravicius). Birkhäuser (2002): 273-297.

 

[11] T. Hara and G. Slade. Mean-field critical behaviour for percolation in high dimensions. Commun. Math. Phys., 128 (1990):333--391.

 

[12] T. Hara and G. Slade. Self-avoiding walk in five or more dimensions. I.  The critical behaviour. Commun. Math. Phys., 147 (1992):101--136.

 

[13] T. Hara and G. Slade. The scaling limit of the incipient infinite cluster in high-dimensional percolation. I. Critical exponents. J. Statist. Phys. 99 (2000): 1075--1168.

 

[14] T. Hara and G. Slade. The scaling limit of the incipient infinite cluster in high-dimensional percolation. II. Integrated super-Brownian excursion. J. Math. Phys. 41 (2000): 1244--1293.

 

[15] R. van der Hofstad, F. den Hollander and G. Slade. A new inductive approach to the lace expansion for self-avoiding walks. Probab. Th. Rel. Fields 111 (1998): 253--286.

 

[16] R. van der Hofstad, F. den Hollander and G. Slade. Construction of the incipient infinite cluster for the spread-out oriented percolation above 4+1 dimensions. Commun. Math. Phys. 231 (2002): 435--461.

 

[17] R. van der Hofstad and A. Sakai. Critical points for spread-out self-avoiding walk, percolation and the contact process above the upper critical dimensions. Preprint (2004).

 

[18] R. van der Hofstad and A. Sakai. Convergence of the critical finite-range contact process to super-Brownian motion above 4 spatial dimensions. In preparation.

 

[19] R. van der Hofstad and G. Slade. A generalised inductive approach to the lace expansion. Probab. Th. Rel. Fields 122 (2002): 389--430.

 

[20] 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 (2003): 413--485.

 

[21] R. van der Hofstad and G. Slade. The lace expansion on a tree with application to networks of self-avoiding walks. Adv. Appl. Math. 30 (2003): 471--528.

 

[22] T. Liggett. Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Springer, Berlin (1999).

 

[23] N. Madras and G. Slade. The Self-Avoiding Walk. Birkhäuser, Boston (1993).

 

[24] B. G. Nguyen and W.-S. Yang. Triangle condition for oriented percolation in high dimensions. Ann. Probab. 21 (1993): 1809--1844.

 

[25] B. G. Nguyen and W.-S. Yang. Gaussian limit for critical oriented percolation in high dimensions. J. Statist. Phys. 78 (1995): 841--876.

 

[26] A. Sakai. Analyses of the critical behavior for the contact process based on a percolation structure.  Ph.D. thesis (2000).

 

[27] A. Sakai. Mean-field critical behavior for the contact process. J. Statist. Phys. 104 (2001): 111--143.

 

[28] A. Sakai. Hyperscaling inequalities for the contact process and oriented percolation. J. Statist. Phys. 106 (2002): 201--211.

 

[29] R. Schonmann. The triangle condition for contact processes on homogeneous trees. J. Statist. Phys. 90 (1998): 1429--1440.

 

[30] G. Slade. The diffusion of self-avoiding random walk in high dimensions. Commun. Math. Phys. 110 (1987): 661-683.

 

[31] G. Slade. Convergence of self-avoiding random walk to Brownian motion in high dimensions. J. Phys. A:  Math. Gen., 21 (1988):L417-L420.

 

[32] G. Slade.  The lace expansion and the upper critical dimension for percolation. Lectures in Applied Mathematics 27 (1991):53--63.

 

[33] G. Slade. The lace expansion and its applications. Saint-Flour lecture notes.  Preprint (2004).

 

[34] C. C. Wu. The contact process on a tree: Behavior near the first phase transition. Stochastic Process. Appl. 57 (1995): 99--112.