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 ${\Re }^d,d\ge 2$. Beginning with a point $X_1$ located at the origin, we generate a random sequence of points $X_1,X_2,\dots ,X_i,\dots ,$. To generate $X_i,i\ge 2$ we choose a uniform integer $j$ in $[i-1]=\{1,2,\dots ,i-1\}$ and then let $X_i=X_j+D_i$ where $D_i=({\delta }_1,\dots ,{\delta }_d)$. 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 $\beta =\min (d,1/\alpha )$.