Convergence of Lattice Trees to Super-Brownian Motion above the Critical Dimension

Mark P. Holmes (U. Auckland)

Abstract


We use the lace expansion to prove asymptotic formulae for the Fourier transforms of the $r$-point functions for a spread-out model of critically weighted lattice trees on the $d$-dimensional integer lattice for $d > 8$. A lattice tree containing the origin defines a sequence of measures on the lattice, and the statistical mechanics literature gives rise to a natural probability measure on the collection of such lattice trees. Under this probability measure, our results, together with the appropriate limiting behaviour for the survival probability, imply convergence to super-Brownian excursion in the sense of finite-dimensional distributions.

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Pages: 671-755

Publication Date: April 18, 2008

DOI: 10.1214/EJP.v13-499

References

  1. R. Adler. Superprocess local and intersection local times and their corresponding particle pictures. In Seminar on Stochastic Processes 1992. Birkhauser, Boston, 1993. MR1278075
  2. D. Aldous. Tree-based models for random distribution of mass. J. Stat. Phys., 73:625--641, 1993. MR1251658
  3. D. Brydges and J. Imbrie. Dimensional reduction formulas for branched polymer correlation functions. J. Stat. Phys., 110:503--518, 2003. MR1964682
  4. D. Brydges and T. Spencer. Self-avoiding walk in 5 or more dimensions. Commun. Math. Phys., 97:125--148, 1985. MR0782962
  5. D. Dawson, I. Iscoe, and E. Perkins. Super-Brownian motion: path properties and hitting probabilities. Probab. Theory Relat. Fields., 83:135--205, 1989. MR1012498
  6. E. Derbez and G. Slade. The scaling limit of lattice trees in high dimensions. Commun. Math. Phys., 193:69--104, 1998. MR1620301
  7. J. Frohlich. Mathematical aspects of the physics of disordered systems. In Phenomenes critiques, systemes aleatoires, theories de jauge, Part II. North-Holland, Amsterdam, 1986. MR0880538
  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. MR1959796
  9. T. Hara and G. Slade. On the upper critical dimension of lattice trees and lattice animals. J. Stat. Phys., 59:1469--1510, 1990. MR1063208
  10. T. Hara and G. Slade. The number and size of branched polymers in high dimensions. J. Stat. Phys., 67:1009--1038, 1992. MR1170084
  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. MR1773141
  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. MR1757958
  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. MR2299712
  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. Poincare Probab. Statist., 43:509--570, 2007. MR2347096
  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. MR1892852
  18. R. van der Hofstad and G. Slade. Convergence of critical oriented percolation to super-Brownian motion above 4+1 dimensions. Ann. Inst. H. Poincar'e Probab. Statist., 39(3):413--485, 2003. MR1978987
  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. MR1973954
  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. MR2031124
  21. M. Holmes and E. Perkins. Weak convergence of measure-valued processes and r-point functions. Ann. Probab., 35:1769--1782, 2007. MR2349574
  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. Birkhauser, Boston, 1993. MR1197356
  25. E. Perkins. Dawson-Watanabe superprocesses and measure-valued diffusions. In Lectures on Probability Theory and Statistics, no. 1781, Ecole d'Ete de Probabilites de Saint Flour 1999. Springer, Berlin, 2002. MR1915445
  26. G. Slade. The lace expansion and its applications, volume 1879 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2006. MR2239599
  27. W. Werner. Random planar curves and Schramm-Loewner evolutions. In Lectures on Probability Theory and Statistics, no. 1840, Ecole d'Ete de Probabilites de Saint Flour 2002. Springer, Berlin, 2004. MR2079672
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