Internal diffusion-limited aggregation with uniform starting points

Abstract

We study internal diffusion-limited aggregation with random starting points on Z d . In this model, each new particle starts from a vertex chosen uniformly at random on the existing aggregate. We prove that the limiting shape of the aggregate is a Euclidean ball. arXiv:1707.03241v1 [math.PR]

IDLA aggregates with infinitely many sources and the directed IDLA forest

“…, nM , let us call particle i the i -th particle which is sent from the interval J M ,k . The main idea to prove Lemma 3.2 is to use the fact that, if particle i hits the strip Z M then, according to Equation (7), it necessarily has to cross a large number of good annuli. But, as we will see in Lemma 3.4, for each new good annulus that particle i meets, it has probability at least η to be stuck inside.…”

“…If one of the nM particles starting from J M ,k hits the strip Z M then the sum of X i 's is necessarily larger than N good . Thus, according to (7), we have…”

“…2(M α + kM ) + 1 and thus N good can be bounded in the same spirit as (7). Now, to control the number of particles starting from J M ,k , let…”

“…Theorem 6.1 is classical in the sense that many results dealing with the shape of aggregates were established for various IDLA models, see e.g. [1,2,7,28], but these results cannot be applied in our context. The proof of Theorem 6.1 relies on an adaptation of [1,2].…”

“…Recently, many variants of this problem have been considered. In particular, IDLA on discrete groups with polynomial or exponential growth have been studied in [5,10], on non-amenable graphs in [18], with multiple sources in [27], on supercritical percolation clusters in [15,33], on comb lattices in [4,19], on cylinder graphs in [23,28,34], constructed with drifted random walks in [29] or with uniform starting points in [7]. A random infinite tree T ∞ can be associated with the sequence of IDLA aggregates (A n ) n≥0 defined above in a very natural way.…”