Aspects of a randomly growing cluster in $\reals^d,d\geq 2

TL;DR

This paper studies a model of a randomly growing cluster in Euclidean space, showing that the cluster remains bounded and exhibits a fractal-like structure with a dimension depending on the growth parameter.

Contribution

It introduces a new probabilistic model of cluster growth in rom which the geometric and fractal properties of the resulting point set are rigorously analyzed.

Findings

The point set is almost surely bounded for all ta>0.

The generated points resemble a eta-dimensional subset of rom a combinatorial perspective.

The Hausdorff dimension of the cluster is eta, where eta= if lpha , else eta=1/lpha.

Abstract

We consider a simple model of a growing cluster of points in $\Re^d,d\geq 2$. Beginning with a point $X_1$ located at the origin, we generate a random sequence of points $X_1,X_2,\ldots,X_i,\ldots,$. To generate $X_{i},i\geq 2$ we choose a uniform integer $j$ in $[i-1]=\{1,2,\ldots,i-1\}$ and then let $X_{i}=X_j+D_i$ where $D_i=(\delta_1,\ldots,\delta_d)$. Here the $\delta_j$ are independent copies of the Normal distribution $N(0,\sigma_i)$, where $\sigma_i=i^{-\alpha}$ for some $\alpha>0$. We prove that for any $\alpha>0$ the resulting point set is bounded a.s., and moreover, that the points generated look like samples from a $\beta$-dimensional subset of $\Re^d$ from the standpoint of the minimum lengths of combinatorial structures on the point-sets, where $\beta=\min(d,1/\alpha)$.