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 Blachère, 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 $α$-stable law, $1 < α<2$, we prove that the cluster contains a contiguous block of $δm +o(m)$ sites, where $0 < δ< 1$, but, unlike the finite-variance case, one may not take $δ=1$.

💡 Research Summary

The paper investigates internal diffusion‑limited aggregation (IDLA) on the one‑dimensional integer lattice ℤ when the driving random walk is allowed to have long‑range jumps. Each walker starts at the origin, performs an i.i.d. step‑distribution X, and stops at the first site that lies outside the current cluster; that site is then added to the cluster. The authors assume a non‑degeneracy (irreducibility) condition guaranteeing that the walk can reach any integer, and they focus on two regimes for the increment distribution X.

**Finite‑variance regime (E