Convergence of lattice trees to super-brownian motion above the critical dimension

Electronic Journal of Probability

Volumn 13, Issue , 2008, Pages 671-755

  Holmes, Mark ^a  

^a UNIVERSITY OF AUCKLAND   (New Zealand)

Author keywords

Lace expansion; Lattice trees; Super Brownian motion

Indexed keywords

EID: 43449097209     PISSN: None     EISSN: 10836489     Source Type: Journal    
DOI: 10.1214/EJP.v13-499     Document Type: Article

References (27)

1. R. Adler. Superprocess local and intersection local times and their corresponding particle pictures. In Seminar on Stochastic Processes 1992. Birkhäuser, Boston, 1993.
2. D. Aldous. Tree-based models for random distribution of mass. J. Stat. Phys., 73:625-641, 1993.
3. D. Brydges and J. Imbrie. Dimensional reduction formulas for branched polymer correlation functions. J. Stat. Phys., 110:503-518, 2003.
4. D. Brydges and T. Spencer. Self-avoiding walk in 5 or more dimensions. Commun. Math. Phys., 97:125-148, 1985.
5. D. Dawson, I. Iscoe, and E. Perkins. Super-Brownian motion: path properties and hitting probabilities. Probab. Theory Relat. Fields., 83:135-205, 1989.
6. E. Derbez and G. Slade. The scaling limit of lattice trees in high dimensions. Commun. Math. Phys., 193:69-104, 1998.
7. J. Frählich. Mathematical aspects of the physics of disordered systems. In Phénomènes critiques, systèmes aléatoires, theories de jauge, Part II. North-Holland, Amsterdam, 1986.
8. T. Hara, R. van der Hofstad, and G. Slade. Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models. Ann. Probab., 31:349-408, 2003.
9. T. Hara and G. Slade. On the upper critical dimension of lattice trees and lattice animals. J. Stat. Phys., 59:1469-1510, 1990.
10. T. Hara and G. Slade. The number and size of branched polymers in high dimensions. J. Stat. Phys., 67:1009-1038, 1992.
11. T. Hara and G. Slade. The scaling limit of the incipient infinite cluster in high-dimensional percolation. i. critical exponents. J. Stat. Phys., 99:1075-1168, 2000.
12. 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., 41:1244-1293, 2000.
13. R. van der Hofstad, F. den Hollander, and G. Slade. The survival probability for critical spread-out oriented percolation above 4+1 dimensions. I. Induction. Probab. Theory Relat. Fields., 138:363-389, 2007.
14. R. van der Hofstad, F. den Hollander, and G. Slade. The survival probability for critical spread-out oriented percolation above 4+1 dimensions. II. Expansion. Ann. Inst. H. Poincaré Probab. Statist., 43:509-570, 2007.
15. R. van der Hofstad, M. Holmes, and G. Slade. An extension of the generalised inductive approach to the lace expansion. Preprint, 2007.
16. 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, 2007.
17. R. van der Hofstad and G. Slade. A generalised inductive approach to the lace expansion. Probab. Theory Relat. Fields., 122:389-430, 2002.
18. R. van der Hofstad and G. Slade. Convergence of critical oriented percolation to superBrownian motion above 4+1 dimensions. Ann. Inst. H. Poincaré Probab. Statist., 39(3):413-485, 2003.
19. R. van der Hofstad and G. Slade. The lace expansion on a tree with application to networks of self-avoiding walks. Adv. Appl. Math., 30:471-528, 2003.
20. M. Holmes, A. Jarai, A. Sakai, and G. Slade. High-dimensional graphical networks of self-avoiding walks. Canadian Journal Math., 56:77-114, 2004.
21. M. Holmes and E. Perkins. Weak convergence of measure-valued processes and r-point functions. Ann. Probab., 35:1769-1782, 2007.
22. J. Klein. Rigorous results for branched polymer models with excluded volume. J. Chem. Phys, 75:5186-5189, 1981.
23. T. Lubensky and J. Isaacson. Statistics of lattice animals and dilute branched polymers. Phys. Rev., A20:2130-2146, 1979.
24. N. Madras and G. Slade. The Self-Avoiding Walk. Birkhäuser, Boston, 1993.
25. E. Perkins. Dawson-Watanabe superprocesses and measure-valued diffusions. In Lectures on Probability Theory and Statistics, no. 1781, Ecole d'Eté de Probabilités de Saint Flour 1999. Springer, Berlin, 2002.
26. G. Slade. The lace expansion and its applications, volume 1879 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2006.
27. W. Werner. Random planar curves and Schramm-Loewner evolutions. In Lectures on Probability Theory and Statistics, no. 1840, Ecole d'Eté de Probabilités de Saint Flour 2002. Springer, Berlin, 2004.