Aspects of a randomly growing cluster in $\reals^d,d\geq 2
Alan Frieze, Ravi Kannan, Wesley Pegden
TL;DR
This work introduces a random growing cluster in $\mathbb{R}^d$ where each new point $X_i$ is obtained by adding a Gaussian increment with variance $\sigma_i=i^{-\alpha}$ to a uniformly chosen earlier point, and establishes that the cluster is almost surely bounded for every $\alpha>0$ while the coarse-dimensional behavior, as seen through minimum spanning tree lengths, corresponds to a dimension $\beta=\min(d,1/\alpha)$. The authors prove an exponential tail for the cluster diameter and analyze the Euclidean MST length $L_n$, deriving upper bounds of $O(n^{1-1/d})$ (with polylog factors for $\alpha\ge 1/d$) and matching lower bounds that depend on the relation between $\alpha$ and $d$, including the special case $\alpha=1/d$. A Polya–Eggenberger urn coupling is used to bound subtree sizes, aiding the lower-bound analysis. The results illuminate how a simple growth mechanism can produce bounded clusters with dimension-like MST scaling, and suggest directions for removing polylog factors and exploring extreme cases such as $\alpha=0$.
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 $D_i=(δ_1,\ldots,δ_d)$. 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 $β=\min(d,1/α)$.