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

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 $\beta=\min(d,1/\alpha)$.