Aspects of a Randomly Growing Cluster in ℝ^2, d≥2

We consider a simple model of a growing cluster of points in \(\mathbb{R}^d, d \geq 2\). 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 \(\mathbb{R}^d\) from the standpoint of the minimum lengths of combinatorial structures on the point-sets, where \(\beta = \min(d,1/\alpha)\).