Abstract
We try to prove rigorously that the perimeter of the large Witten-Sander cluster does not scale as the square root of its area, by making a forced comparison with the ill-posed Hele-Shaw problem of fluid dynamics. The attempt is not completely successful; nevertheless some interesting consequences of the comparison are derived.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
D. Bensimonet al., Viscous flows in two dimensions,Rev. Mod. Phys. 58:977–990 (1986).
C. Caratheodory,Conformal Representation, 2nd ed. (Cambridge University Press, Cambridge, 1958).
A. De Masi, P. A. Ferrari, and J. L. Lebowitz, Reaction-diffusion equations for interacting particle systems,J. Stat. Phys. 44:589–644 (1986).
A. De Masi, P. A. Ferrari, and M. E. Vares, A microscopic model of interface related to the Burger's equation,J. Stat. Phys. 55:601–611 (1989).
E. Di Benedetto and A. Friedman, The ill-posed Hele-Shaw model and the Stefan problem for supercooled water,Trans. Am. Math. Soc. 282:183–204 (1984).
J. L. Doob,Classical Potential Theory and its Probabilistic Counterpart (Springer-Verlag, New York, 1984).
A. Friedman,Variational Problems and Free Boundary Problems (Wiley, New York, 1982).
A. Joffe and M. Metivier, Weak convergence of sequences of semimartingales with applications to multiple branching processes,Adv. Appl. Prob. 18:20–65 (1986).
L. Kadanoff, Simulating hydrodynamics: A pedestrian model,J. Stat. Phys. 39:267–283 (1985).
O. D. Kellogg,Foundations of Potential Theory (Verlag von J. Springer, Berlin, 1929).
H. Kesten, How long are the arms in DLA?,J. Phys. A: Math. Gen. 20:L29-L33 (1987).
H. Kesten, Hitting probabilities of random walks onℤ d,Stock. Proc. Appl. 25:165–184 (1987).
H. Kesten, Some caricatures of multiple contact diffusion limited aggregation and theη-model, inProceedings of Durham Symposium on Stochastic Analysis, to appear.
C. Kipnis, S. Olla, and S. R. S. Varadhan, Hydrodynamics and large deviation for simple exclusion processes,Commun. Pure Appl. Math. 42:115–137 (1989).
S. Liang, Random walk simulations of flow in Hele-Shaw cells,Phys. Rev. A 33:2663–2674 (1986).
I. Mitoma, Tightness of probabilities onC([0, 1], ℒ′) andD([0, 1], ℒ′), Ann. Prob. 11:989–999 (1983).
C. Pommerenke,Univalent Functions (Vanderhoeck and Ruprecht, 1976).
L. Paterson, Diffusion limited aggregation and two fluid displacement in porous media,Phys. Rev. Lett. 53:1621–1624 (1984).
M. Plischke, Z. Rácz, and D. Liu, Time reversal invariance and universality of two dimensional growth models,Phys. Rev. B 35:3485–3495 (1987).
F. Spitzer,Principles of Random Walk (Van Nostrand, New York).
H. E. Stanley and N. Ostrowsky,On Growth and Form: Fractal and Non-fractal Patterns in Physics (M. Nijhoff, Boston, 1986).
H. E. Stanley and N. Ostrowsky,Nato Advanced Study Institute on Random Fluctuations and Pattern Growth: Experiments and Models (Kluwer Academic Publishers, Boston, 1988).
E. M. Stein,Singular Integrals and Differentiability Properties of Functions (Princeton University Press, Princeton, New Jersey, 1970).
D., Stroock and S. R. S. Varadhan,Multidimensional Diffusion Processes (Springer-Verlag, Berlin, 1979).
J. Szep, J. Cserti, and J. Kertész, Monte-Carlo approach to dendritic growth,J. Phys. A 18:L413–4l8 (1985).
T. Viczek,Fractal Growth Phenomena (World Scientific, Singapore, 1989).
T. A. Witten and L. M. Sander, Diffusion limited aggregation, a kinetic critical phenomenon,Phys. Rev. Lett. 47:1400–1403 (1981).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
March, P. Remarks on scaling a model of Witten-Sander type. J Stat Phys 67, 1117–1149 (1992). https://doi.org/10.1007/BF01049012
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01049012