We consider a cluster growth model on Z^d, called internal diffusion limited aggregation (internal DLA). In this model, random walks start at the origin, one at a time, and stop moving when reaching a site not occupied by previous walks. It is known that the asymptotic shape of the cluster is spherical. Also, when dimension is 2 or more, and when the cluster has volume $n^d$, it is known that fluctuations of the radius are at most of order $n^{1/3}$. We improve this estimate to $n^{1/(d+1)}$, in dimension 3 or more. In so doing, we introduce a closely related cluster growth model, that we call the flashing process, whose fluctuations are controlled easily and accurately. This process is coupled to internal DLA to yield the desired bound.
Part of our proof adapts the approach of Lawler, Bramson and Griffeath, on another space scale, and uses a sharp estimate (written by Blachere in our Appendix) on the expected time spent by a random walk inside an annulus.
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.
Internal diffusion limited aggregation (IDLA) is a random aggregation model on a graph $G$, whose clusters are formed by random walks started in the origin (some fixed vertex) and stopped upon visiting a previously unvisited site. On the Sierpinski gasket graph the asymptotic shape is known to be a ball in the usual graph metric. In this paper we establish bounds for the fluctuations of the cluster from its asymptotic shape.
我们研究了整数格Z^d上三种不同聚集模型的标度极限：内部DLA，其中粒子执行随机游动，直到到达一个未被占用的站点;转子路由器模型，其中粒子执行确定性类似的随机游动;和可分的沙堆，其中每个站点将其多余的质量均匀地分配给其邻居。当晶格间距趋于零时，我们发现三种模型具有相同的尺度极限，我们将其描述为R^d中某个PDE自由边界问题的解。特别是，内部DLA具有确定性的缩放限制。我们发现标度极限是在位势论和流体力学等许多领域独立出现的求积域。我们的结果既适用于重点源的情况，也适用于Diaconis-Fulton smash和域。在特殊情况下，当所有的粒子开始在一个单一的网站，我们表明，标度限制是一个欧几里德球在R^d，并给出了定量的界限上的收敛速率的球。对于经典阿贝尔沙堆模型的形状，我们还改进了以前最著名的Le Borgne和Rossin在Z^2中的边界以及Fey和Redig在高维中的边界。最后，我们研究了叶子被折叠到单个汇点的正则树的沙堆群，并确定了满沙堆群的分解为循环群的乘积。例如，对于高度为n的正则三元树，沙堆群同构于（Z_3）^{2^{n-3}} x（Z_7）^{2^{n-4}} x. x Z_{2^{n-1}-1} x Z_{2^n-1}。我们用这个结果来证明，转子路由器聚集在定期树产生一个完美的球。
Internal DLA (IDLA) is an internal aggregation model in which particles perform random walks from the origin, in turn, and stop upon reaching an unoccupied site. Levine and Peres showed that, when particles start instead from fixed multiple-point distributions, the modified IDLA processes have deterministic scaling limits related to a certain obstacle problem. In this paper, we investigate the convergence rate of this "extended source" IDLA in the plane to its scaling limit. We show that, if $δ$ is the lattice size, fluctuations of the IDLA occupied set are at most of order $δ^{3/5}$ from its …
We had established inner and outer fluctuation for the internal DLA cluster when all walks are launched from the origin. In obtaining the outer fluctuation, we had used a deep lemma of Jerison, Levine and Sheffield, which estimate roughly the possibility of fingering, and had provided a simple proof using an interesting estimate for crossing probability for a simple random walk. The application of the crossing probability to the fingering for the internal DLA cluster contains a flaw discovered recently, that we correct in this note. We take the opportunity to make a self-contained exposition.
We show that Internal Diffusion Limited Aggregation (IDLA) on $\mathbb{Z}^d$ has near optimal Cheeger constant when the growing cluster is large enough. This implies, through a heat kernel lower bound derived previously in [H], that simple random walk evolving independently on growing in time IDLA cluster is recurrent when $d\ge 3$.
Internal diffusion-limited aggregation (IDLA) is a stochastic growth model on a graph $G$ which describes the formation of a random set of vertices growing from the origin (some fixed vertex) of $G$. Particles start at the origin and perform simple random walks; each particle moves until it lands on a site which was not previously visited by other particles. This random set of occupied sites in $G$ is called the IDLA cluster. In this paper we consider IDLA on Sierpinski gasket graphs, and show that the IDLA cluster fills balls (in the graph metric) with probability 1.
