Convergence of critical oriented percolation to super-brownian motion above 4+1 dimensions
Van der Hofstad, Remco ; Slade, Gordon
Annales de l'I.H.P. Probabilités et statistiques, Tome 39 (2003), p. 413-485 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU) 
@article{AIHPB_2003__39_3_413_0,
     author = {Van der Hofstad, Remco and Slade, Gordon},
     title = {Convergence of critical oriented percolation to super-brownian motion above $4+1$ dimensions},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     volume = {39},
     year = {2003},
     pages = {413-485},
     doi = {10.1016/S0246-0203(03)00008-6},
     mrnumber = {1978987},
     zbl = {1020.60099},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPB_2003__39_3_413_0}
}

Van der Hofstad, Remco; Slade, Gordon. Convergence of critical oriented percolation to super-brownian motion above $4+1$ dimensions. Annales de l'I.H.P. Probabilités et statistiques, Tome 39 (2003) pp. 413-485. doi : 10.1016/S0246-0203(03)00008-6. http://gdmltest.u-ga.fr/item/AIHPB_2003__39_3_413_0/

[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. | MR 1278075 | Zbl 0786.60103

[2] M. Aizenman, D.J. Barsky, Sharpness of the phase transition in percolation models, Comm. Math. Phys. 108 (1987) 489-526. | MR 874906 | Zbl 0618.60098

[3] M. Aizenman, C.M. Newman, Tree graph inequalities and critical behavior in percolation models, J. Stat. Phys. 36 (1984) 107-143. | MR 762034 | Zbl 0586.60096

[4] D. Aldous, Tree-based models for random distribution of mass, J. Stat. Phys. 73 (1993) 625-641. | MR 1251658 | Zbl 1102.60318

[5] M.T. Barlow, E.A. Perkins, On the filtration of historical Brownian motion, Ann. Probab. 22 (1994) 1273-1294. | MR 1303645 | Zbl 0816.60044

[6] C. Bezuidenhout, G. Grimmett, The critical contact process dies out, Ann. Probab. 18 (1990) 1462-1482. | MR 1071804 | Zbl 0718.60109

[7] E. Bolthausen, C. Ritzmann, A central limit theorem for convolution equations and weakly self-avoiding walks, Ann. Probab., to appear.

[8] D.C. Brydges, T. Spencer, Self-avoiding walk in 5 or more dimensions, Comm. Math. Phys. 97 (1985) 125-148. | MR 782962 | Zbl 0575.60099

[9] J.T. Cox, R. Durrett, E.A. Perkins, Rescaled voter models converge to super-Brownian motion, Ann. Probab. 28 (2000) 185-234. | MR 1756003 | Zbl 1044.60092

[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. | MR 1703136 | Zbl 0941.60095

[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. | MR 1242575 | Zbl 0799.60080

[12] E. Derbez, G. Slade, Lattice trees and super-Brownian motion, Canad. Math. Bull. 40 (1997) 19-38. | MR 1443722 | Zbl 0874.60091

[13] E. Derbez, G. Slade, The scaling limit of lattice trees in high dimensions, Comm. Math. Phys. 193 (1998) 69-104. | MR 1620301 | Zbl 0915.60076

[14] R. Durrett, E.A. Perkins, Rescaled contact processes converge to super-Brownian motion in two or more dimensions, Probab. Theory Related Fields 114 (1999) 309-399. | MR 1705115 | Zbl 0953.60093

[15] E.B. Dynkin, Representation for functionals of superprocesses by multiple stochastic integrals, with applications to self-intersection local times, Astérisque 157-158 (1988) 147-171. | Zbl 0659.60105

[16] E.B. Dynkin, An Introduction to Branching Measure-Valued Processes, American Mathematical Society, Providence, RI, 1994. | MR 1280712 | Zbl 0824.60001

[17] A.M. Etheridge, An Introduction to Superprocesses, American Mathematical Society, Providence, RI, 2000. | MR 1779100 | Zbl 0971.60053

[18] G. Grimmett, Percolation, Springer, Berlin, 1999. | MR 1707339 | Zbl 0926.60004

[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. | MR 1901958 | Zbl 1010.60087

[20] T. Hara, G. Slade, Mean-field critical behaviour for percolation in high dimensions, Comm. Math. Phys. 128 (1990) 333-391. | MR 1043524 | Zbl 0698.60100

[21] T. Hara, G. Slade, The number and size of branched polymers in high dimensions, J. Stat. Phys. 67 (1992) 1009-1038. | MR 1170084 | Zbl 0925.82231

[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. | MR 1773141 | Zbl 0968.82016

[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. | MR 1757958 | Zbl 0977.82022

[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. | MR 1946445 | Zbl 1013.82017

[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 Fields 111 (1998) 253-286. | MR 1633582 | Zbl 0906.60078

[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. | MR 1973954 | Zbl 1027.60098

[27] R. Van Der Hofstad, G. Slade, A generalised inductive approach to the lace expansion, Probab. Theory Related Fields 122 (2002) 389-430. | MR 1892852 | Zbl 1002.60095

[28] J.-F. Le Gall, Spatial Branching Processes, Random Snakes, and Partial Differential Equations, Birkhäuser, Basel, 1999. | MR 1714707 | Zbl 0938.60003

[29] N. Madras, G. Slade, The Self-Avoiding Walk, Birkhäuser, Boston, 1993. | MR 1197356 | Zbl 0872.60076

[30] M.V. Menshikov, Coincidence of critical points in percolation problems, Soviet Math. Dokl. 33 (1986) 856-859. | MR 852458 | Zbl 0615.60096

[31] B.G. Nguyen, W.-S. Yang, Triangle condition for oriented percolation in high dimensions, Ann. Probab. 21 (1993) 1809-1844. | MR 1245291 | Zbl 0806.60097

[32] B.G. Nguyen, W.S. Yang, Gaussian limit for critical oriented percolation in high dimensions, J. Stat. Phys. 78 (1995) 841-876. | MR 1315235 | Zbl 01553900

[33] S.P. Obukhov, The problem of directed percolation, Phys. 101A (1980) 145-155. | MR 572004

[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. | Zbl 1020.60075

[35] A. Sakai, Mean-field critical behavior for the contact process, J. Stat. Phys. 104 (2001) 111-143. | MR 1851386 | Zbl 1019.82012

[36] A. Sakai, Hyperscaling inequalities for the contact process and oriented percolation, J. Stat. Phys. 106 (2002) 201-211. | MR 1881726 | Zbl 0993.82024

[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. | MR 1703123 | Zbl 0942.60072

[38] G. Slade, Scaling limits and super-Brownian motion, Notices Amer. Math. Soc. 49 (9) (2002) 1056-1067. | MR 1927455 | Zbl 02115022