# zbMATH — the first resource for mathematics

Stabilization of DLA in a wedge. (English) Zbl 1441.60087
Summary: We consider diffusion limited aggregation (DLA) in a two-dimensional wedge. We prove that if the angle of the wedge is smaller than $$\pi/4$$, there is some $$a>2$$ such that almost surely, for all $$R$$ large enough, after time $$R^a$$ all new particles attached to the DLA will be at distance larger than $$R$$ from the origin. Furthermore, we provide estimates on the size of $$R$$ under which this holds. This means that DLA stabilizes in growing balls, thus allowing a definition of the infinite DLA in a wedge via a finite time process.
##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60K40 Other physical applications of random processes 60G50 Sums of independent random variables; random walks
Full Text:
##### References:
 [1] [BLPS01] I. Benjamini, R. Lyons, Y. Peres, and O. Schramm. Special invited paper: uniform spanning forests. Annals of Probability, 1-65, 2001. · Zbl 1016.60009 [2] [BY17] I. Benjamini and A. Yadin. Upper bounds on the growth rate of diffusion limited aggregation. arXiv preprint arXiv:1705.06095, 2017. [3] [BQ06] K. Burdzy and J. Quastel. An annihilating-branching particle model for the heat equation with average temperature zero. The Annals of Probability, 2382-2405, 2006. · Zbl 1122.60085 [4] [CF17] Z. Chen and W. Fan. Hydrodynamic limits and propagation of chaos for interacting random walks in domains. The Annals of Applied Probability, 27(3):1299-1371, 2017. · Zbl 1372.60133 [5] [GP17] S. Ganguly and Y. Peres. Convergence of discrete Green functions with Neumann boundary conditions. Potential Analysis, 46(4):799-818, 2017. · Zbl 1369.60022 [6] [HMP86] T. C. Halsey, P. Meakin, and I. Procaccia. Scaling structure of the surface layer of diffusion-limited aggregates. Physical Review Letters, 56(8):854, 1986. [7] [HLLS18] N. Holden, G. F. Lawler, X. Li, and X. Sun. Minkowski content of Brownian cut points. arXiv preprint arXiv:1803.10613, 2018. [8] $$[KOO^+98]$$ D. A. Kessler, Z. Olami, J. Oz, I. Procaccia, E. Somfai, and L. M. Sander. Diffusion-limited aggregation and viscous fingering in a wedge: Evidence for a critical angle. Physical Review E, 57(6):6913, 1998. [9] [Kes87a] H. Kesten. Hitting probabilities of random walks on $${\mathbb{Z} }^d$$. Stochastic Process. Appl., 25(2):165-184, 1987. · Zbl 0626.60067 [10] [Kes87b] H. Kesten. How long are the arms in DLA? J. Phys. A, 20(1):L29-L33, 1987. [11] [LL10] G. F. Lawler and V. Limic. Random walk: a modern introduction. Cambridge University Press, 2010. · Zbl 1210.60002 [12] [LSW03] G. Lawler, O. Schramm, and W. Werner. Conformal restriction: the chordal case. Journal of the American Mathematical Society, 16(4):917-955, 2003. · Zbl 1030.60096 [13] [LP17] R. Lyons and Y. Peres. Probability on trees and networks. Cambridge University Press, 2017. [14] [MP10] P. Morters and Y. Peres. Brownian motion, volume 30. Cambridge University Press, 2010. [15] [PZ17] E. B. Procaccia and Y. Zhang. On sets of zero stationary harmonic measure. arXiv preprint arXiv:1711.01013, 2017. [16] [PZ19] Eviatar B. Procaccia and Yuan Zhang. Stationary harmonic measure and DLA in the upper half plane. Journal of Statistical Physics, 176(4):946-980, 2019. · Zbl 07115581 [17] [SV71] D. W. Stroock and S.R.S. Varadhan. Diffusion processes with boundary conditions. Communications on Pure and Applied Mathematics, 24(2):147-225, 1971. · Zbl 0227.76131 [18] [WS83] T. A. Witten and L. M. Sander. Diffusion-limited aggregation. Phys. Rev. B (3), 27(9):5686-5697, 1983.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.