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 $D_i=(δ_1,\ldots,δ_d)$. Here the $δ_j$ are independent copies of the Normal distribution $N(0,σ_i)$, where $σ_i=i^{-α}$ for some $α>0$. We prove that for any $α>0$ the resulting point set is bounded a.s., and moreover, that the points generated look like samples from a $β$-dimensional subset of $\Re^d$ from the standpoint of the minimum lengths of combinatorial structures on the point-sets, where $β=\min(d,1/α)$.