Convergence of critical oriented percolation to super-brownian motion above $4+1$ dimensions

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1. [1] R.J. Adler, Superprocess local and intersection local times and their corresponding particle pictures, in: Çinlar E., Chung K.L., Sharpe M.J. (Eds.), Seminar on Stochastic Processes 1992, Birkhäuser, Boston, 1993, pp. 1-42. Zbl0786.60103MR1278075
2. [2] M. Aizenman, D.J. Barsky, Sharpness of the phase transition in percolation models, Comm. Math. Phys.108 (1987) 489-526. Zbl0618.60098MR874906
3. [3] M. Aizenman, C.M. Newman, Tree graph inequalities and critical behavior in percolation models, J. Stat. Phys.36 (1984) 107-143. Zbl0586.60096MR762034
4. [4] D. Aldous, Tree-based models for random distribution of mass, J. Stat. Phys.73 (1993) 625-641. Zbl1102.60318MR1251658
5. [5] M.T. Barlow, E.A. Perkins, On the filtration of historical Brownian motion, Ann. Probab.22 (1994) 1273-1294. Zbl0816.60044MR1303645
6. [6] C. Bezuidenhout, G. Grimmett, The critical contact process dies out, Ann. Probab.18 (1990) 1462-1482. Zbl0718.60109MR1071804
7. [7] E. Bolthausen, C. Ritzmann, A central limit theorem for convolution equations and weakly self-avoiding walks, Ann. Probab., to appear. Zbl06571500
8. [8] D.C. Brydges, T. Spencer, Self-avoiding walk in 5 or more dimensions, Comm. Math. Phys.97 (1985) 125-148. Zbl0575.60099MR782962
9. [9] J.T. Cox, R. Durrett, E.A. Perkins, Rescaled voter models converge to super-Brownian motion, Ann. Probab.28 (2000) 185-234. Zbl1044.60092MR1756003
10. [10] T. Cox, R. Durrett, E.A. Perkins, Rescaled particle systems converging to super-Brownian motion, in: Bramson M., Durrett R. (Eds.), Perplexing Problems in Probability: Festschrift in Honor of Harry Kesten, Birkhäuser, Basel, 1999. Zbl0941.60095MR1703136
11. [11] D.A. Dawson, Measure-Valued Markov Processes, in: Ecole d'Eté de Probabilités de Saint-Flour 1991, Lecture Notes in Mathematics, 1541, Springer, Berlin, 1993. Zbl0799.60080MR1242575
12. [12] E. Derbez, G. Slade, Lattice trees and super-Brownian motion, Canad. Math. Bull.40 (1997) 19-38. Zbl0874.60091MR1443722
13. [13] E. Derbez, G. Slade, The scaling limit of lattice trees in high dimensions, Comm. Math. Phys.193 (1998) 69-104. Zbl0915.60076MR1620301
14. [14] R. Durrett, E.A. Perkins, Rescaled contact processes converge to super-Brownian motion in two or more dimensions, Probab. Theory Related Fields114 (1999) 309-399. Zbl0953.60093MR1705115
15. [15] E.B. Dynkin, Representation for functionals of superprocesses by multiple stochastic integrals, with applications to self-intersection local times, Astérisque157–158 (1988) 147-171. Zbl0659.60105
16. [16] E.B. Dynkin, An Introduction to Branching Measure-Valued Processes, American Mathematical Society, Providence, RI, 1994. Zbl0824.60001MR1280712
17. [17] A.M. Etheridge, An Introduction to Superprocesses, American Mathematical Society, Providence, RI, 2000. Zbl0971.60053MR1779100
18. [18] G. Grimmett, Percolation, Springer, Berlin, 1999. Zbl0926.60004MR1707339
19. [19] G. Grimmett, P. Hiemer, Directed percolation and random walk, in: Sidoravicius V. (Ed.), In and Out of Equilibrium, Birkhäuser, Boston, 2002, pp. 273-297. Zbl1010.60087MR1901958
20. [20] T. Hara, G. Slade, Mean-field critical behaviour for percolation in high dimensions, Comm. Math. Phys.128 (1990) 333-391. Zbl0698.60100MR1043524
21. [21] T. Hara, G. Slade, The number and size of branched polymers in high dimensions, J. Stat. Phys.67 (1992) 1009-1038. Zbl0925.82231MR1170084
22. [22] T. Hara, G. Slade, The scaling limit of the incipient infinite cluster in high-dimensional percolation. I. Critical exponents, J. Stat. Phys.99 (2000) 1075-1168. Zbl0968.82016MR1773141
23. [23] T. Hara, 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. Zbl0977.82022MR1757958
24. [24] R. van der Hofstad, F. den Hollander, G. Slade, Construction of the incipient infinite cluster for spread-out oriented percolation above 4+1 dimensions, Comm. Math. Phys.231 (2002) 435-461. Zbl1013.82017MR1946445
25. [25] R. van der Hofstad, F. den Hollander, G. Slade, A new inductive approach to the lace expansion for self-avoiding walks, Probab. Theory Related Fields111 (1998) 253-286. Zbl0906.60078MR1633582
26. [26] R. van der Hofstad, G. Slade, The lace expansion on a tree with application to networks of self-avoiding walks, Adv. Appl. Math., to appear. Zbl1027.60098MR1973954
27. [27] R. van der Hofstad, G. Slade, A generalised inductive approach to the lace expansion, Probab. Theory Related Fields122 (2002) 389-430. Zbl1002.60095MR1892852
28. [28] J.-F. Le Gall, Spatial Branching Processes, Random Snakes, and Partial Differential Equations, Birkhäuser, Basel, 1999. Zbl0938.60003MR1714707
29. [29] N. Madras, G. Slade, The Self-Avoiding Walk, Birkhäuser, Boston, 1993. Zbl0872.60076MR1197356
30. [30] M.V. Menshikov, Coincidence of critical points in percolation problems, Soviet Math. Dokl.33 (1986) 856-859. Zbl0615.60096MR852458
31. [31] B.G. Nguyen, W.-S. Yang, Triangle condition for oriented percolation in high dimensions, Ann. Probab.21 (1993) 1809-1844. Zbl0806.60097MR1245291
32. [32] B.G. Nguyen, W.S. Yang, Gaussian limit for critical oriented percolation in high dimensions, J. Stat. Phys.78 (1995) 841-876. Zbl1114.82304MR1315235
33. [33] S.P. Obukhov, The problem of directed percolation, Phys.101A (1980) 145-155. MR572004
34. [34] E. Perkins, Dawson–Watanabe superprocesses and measure-valued diffusions, in: Bernard P.L. (Ed.), Lectures on Probability Theory and Statistics. Ecole d'Eté de Probabilités de Saint-Flour XXIX-1999, Lecture Notes in Mathematics, 1781, Springer, Berlin, 2002, pp. 125-329. Zbl1020.60075
35. [35] A. Sakai, Mean-field critical behavior for the contact process, J. Stat. Phys.104 (2001) 111-143. Zbl1019.82012MR1851386
36. [36] A. Sakai, Hyperscaling inequalities for the contact process and oriented percolation, J. Stat. Phys.106 (2002) 201-211. Zbl0993.82024MR1881726
37. [37] G. Slade, Lattice trees, percolation and super-Brownian motion, in: Bramson M., Durrett R. (Eds.), Perplexing Problems in Probability: Festschrift in Honor of Harry Kesten, Birkhäuser, Basel, 1999. Zbl0942.60072MR1703123
38. [38] G. Slade, Scaling limits and super-Brownian motion, Notices Amer. Math. Soc.49 (9) (2002) 1056-1067. Zbl1126.60322MR1927455

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