Long-range one-dimensional internal diffusion-limited aggregation

We study internal diffusion limited aggregation on $\mathbb{Z}$, where a cluster is grown incrementally by adding, for each random walk dispatched from the origin, the first site it reaches outside the cluster. We assume that the increment distribution $X$ of the driving random walks has $\mathbb{E} X =0$, but need neither be simple nor symmetric, and can have $\mathbb{E} (X^2) = \infty$, for example. For the case where $\mathbb{E} (X^2) < \infty$, we prove that after $m$ of the random walks have been dispatched, all but $o(m)$ sites in the cluster form an approximately symmetric contiguous block around the origin. This strengthens a result of Blach\`ere, for centred random walks whose increments have finite $3$rd moments, to the optimal moments condition. On the other hand, if $X$ is in the domain of attraction of a symmetric $\alpha$-stable law, $1 < \alpha <2$, we prove that the cluster contains a contiguous block of $\delta m +o(m)$ sites, where $0 < \delta < 1$, but, unlike the finite-variance case, one may not take $\delta=1$.