IDLA in wedged like regions: Unaccessible Corners

General Description View the group

Assume we have $n$ identical particles at the origin of the lattice $\mathbb Z^d$ ($d\geq 2$). One by one each of these particles starts a simple symmetric random walk from the origin (with walks being independent of each other), until reaching an unoccupied site (vertex) on $\mathbb Z^d$. Then the particle stops, and never moves from there.

All particles follow this process, until we get at most 1 particle hosted by any vertex of $\mathbb Z^d$. One can prove, that this process, which is called internal diffusion limited aggregation (IDLA for short), with probability 1 will reach a stable state.

Let $D_n\subset \mathbb Z^d$ be the set of occupied vertices of $\mathbb Z^d$ for $n$ particles at the origin. It is proved in [1] that as $n \to \infty$, the set $n^{-1/d} D_n$ converges to a ball with probability 1; see [1] for the precise statement on existence of the scaling limit of IDLA. For more general point sources, see [2].

Here I shall describe two problems related to this model, on wedge shaped domains, where the boundary of the domains play the role of absorbents, and particles reaching there will disappear from the system.

Problem's Description

In the model described before we let $d=2$ and $z=(1,0)$. We shall now consider an IDLA in a wedge shaped domain, such as $$D:=\{ x_1 \geq k |x_2|\} \cap \frac{1}{n} \mathbb Z^2, \qquad \hbox{ for some } k > 0.$$ Governing rules:

i) Initially $D \setminus z$ has no particles.

ii) Particles that reach $\partial D$ are absorbed and disappear from the system.

Question:How many particles $N=N(k,n)$ (in average) do we need to run before we can reach the origin?

The scaling limit of this problem (that one may prove to exist as in [2]) is what is known as quadrature domain and Hele-shaw flow restricted to a domain. For a related problem in continuous setting see [3].

1. ArticleIs an originInternal diffusion limited aggregation.

   • Maury Bramson,
   • David Griffeath,
   • Gregory F. Lawler

   Annals of Probability 20 (4), 2177--2140, 1992

2. Thesis Limit Theorems for Internal Aggregation Models,

   • Lionel Levine

   PhD thesis, University of California Berkley, 2007

3. ArticleIs an originWhen does the free boundary enter into corner points of the fixed boundary?

   • Henrik Shahgholian

   Zapiski Nauchnyh Seminarov POMI 310, 2004

4. ArticleIs an originSmall modifications of quadrature domains.

   • Makoto Sakai

   Mem. Amer. Math. Soc. 206 (969), 2010