Aspects of a randomly growing cluster in R d , d ≥ 2

We consider a simple model of a growing cluster of points in R d , d ≥ 2 . Beginning with a point X 1 located at the origin, we generate a random sequence of points X 1 , X 2 , … , X i , … , . To generate X i , i ≥ 2 we choose a uniform integer j in [ i − 1 ] = 1 , 2 , … , i − 1 and then let X i = X j + D i where D i = ( δ 1 , … , 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 R d from the standpoint of the minimum lengths of combinatorial structures on the point-sets, where β = min ( d , 1 / α ) .