PR ] 5 A pr 2 00 4 DIFFUSION-LIMITED AGGREGATION ON A TREE by

We study the following growth model on a regular d-ary tree. Points at distance n adjacent to the existing subtree are added with probabilities proportional to α, where α < 1 is a positive real parameter. The heights of these clusters are shown to increase linearly with their total size; this complements known results that show the height increases only logarithmically when α ≥ 1. Results are obtained using stochastic monotonicity and regeneration results which may be of independent interest. Our motivation comes from two other ways in which the model may be viewed: as a problem in first-passage percolation, and as a version of diffusion-limited aggregation (DLA), adjusted so that “fingering” occurs.

[1]  Parkpoom Phetpradap,et al.  Intersections of random walks , 2011 .

[2]  Aarnout Brombacher,et al.  Probability... , 2009, Qual. Reliab. Eng. Int..

[3]  Stefan Grosskinsky Warwick,et al.  Interacting Particle Systems , 2016 .

[4]  E. Rio.  The Functional Law of the Iterated Logarithm for Stationary Strongly Mixing Sequences , 1995 .

[5]  R. Pemantle,et al.  Domination Between Trees and Application to an Explosion Problem , 2004, math/0404044.

[6]  M. Barlow.  Fractals, and diffusion-limited aggregation , 1993 .

[7]  Hosam M. Mahmoud,et al.  Evolution of random search trees , 1991, Wiley-Interscience series in discrete mathematics and optimization.

[8]  R. Durrett.  Probability: Theory and Examples , 1993 .

[9]  H. Kesten.  Upper bounds for the growth rate of DLA , 1990 .

[10]  T. Vicsek.  Fractal Growth Phenomena , 1989 .

[11]  D. Aldous,et al.  A diffusion limit for a class of randomly-growing binary trees , 1988 .

[12]  H. Kesten.  How long are the arms in DLA , 1987 .

[13]  Luc Devroye,et al.  A note on the height of binary search trees , 1986, JACM.

[14]  B. Pittel.  Asymptotical Growth of a Class of Random Trees , 1985 .

[15]  K. Athreya.  Discounted branching random walks , 1985, Advances in Applied Probability.

[16]  B. Pittel.  On growing random binary trees , 1984 .

[17]  J. Vannimenus,et al.  Models of cluster growth on the Cayley tree , 1984 .

[18]  Charles M. Newman,et al.  Normal fluctuations and the FKG inequalities , 1980 .

[19]  Jacob Ziv,et al.  Coding theorems for individual sequences , 1978, IEEE Trans. Inf. Theory.

[20]  M. Bramson.  Minimal displacement of branching random walk , 1978 .

[21]  T. Kamae,et al.  Stochastic Inequalities on Partially Ordered Spaces , 1977 .

[22]  J. Kingman.  The First Birth Problem for an Age-dependent Branching Process , 1975 .

[23]  G. Lorden.  On Excess Over the Boundary , 1970 .

[24]  F. G. Foster,et al.  An Introduction to Probability Theory and its Applications, Volume I (2Nd Edition) , 1958 .

[25]  Feller William,et al.  An Introduction To Probability Theory And Its Applications , 1950 .