Some pictures in geometric probability

As I discussed in a previous blog post, I have been recently interested in models of randomly growing networks. As a starting point I focused my attention on the preferential attachment rule and its variants, in part because its ubiquity in the literature. However these models miss one key feature, namely the notion of geometry: when a newcomer arrives in the network, she can connect to any vertex, regardless of a possible spatial organization of the network and her current position in that space.

In this blog post I want to discuss some of the most famous (or infamous in some cases) spatial models of network growth. For simplicity, and also to draw pictures, everything will be described in the plane, and specifically on the lattice \mathbb{Z}^2. These models iteratively build an aggregate A_n \subset \mathbb{Z}^2 as follows: First A_0 = \{(0,0)\}. Then A_{n+1}=A_n \cup \{X_n\} where X_n is a random point in the boundary of the aggregate, that is X_n \in \partial A_n := \{x \in \mathbb{Z}^2 : \min_{y \in A_n} \|x-y\|_2 =1\}. Thus a model is defined by the probability measure one puts on the boundary of the aggregate to select the next point to add.

All the pictures are with half a million particles, and the particles are colored as a function of their age, with blue corresponding to old particles, and red corresponding to young ones.

Eden model

The simplest model uses the uniform measure on the boundary, and it is known as the Eden model. Here is a picture:


One of the most basic result about this model is that it admits a limit shape:

Theorem: There exists a (deterministic) convex set B \subset \mathbb{R}^2 such that for any \epsilon >0,

    \[\lim_{n \to +\infty} \mathbb{P} \left( (1- \epsilon) B \subset \frac{1}{\sqrt{n}} A_n \subset (1+\epsilon) B \right) = 1.\]

It is known that B is not an Euclidean ball (this should be clear from the picture), though nobody knows what B is exactly. How do you prove such a theorem? Well it turns out to be pretty easy once you have the right machinery. The first step is to realize that the Eden model exactly corresponds to first passage percolation with exponential clocks: imagine that you have i.i.d. exponential random variables t(e) on the edges of \mathbb{Z}^2, and consider A(t) \subset \mathbb{Z}^2 to be the set of points x such that there exists a path e_1, \hdots e_r from (0,0) to x with \sum_{s=1}^r t(e_s) \leq t. In other words A(t) is the set of points reached at time t by a fluid released at time 0 in (0,0) and with travel times on the edges given by the random variables t(e). It is an easy exercise to convince yourself that A_n has the same distribution as A(t) conditionally on |A(t)| = n (this is actually not true, one in fact needs to put the travel times on the vertices of \mathbb{Z}^2 rather than on the edges, but let me get away with that small mistake). At this point the continuous version of the above theorem can be rather easily proved via Kingman’s subadditive ergodic theorem, see the Saint Flour lecture notes by Kesten for more details.


Diffusion Limited Aggregation (DLA)

Here we consider the harmonic measure from infinity: pick a point of \mathbb{Z}^2 uniformly at random from a large circle that contains A_n, and then start a random walk from this point until it hits \partial A_n (which will happen eventually since a simple random walk in two dimensions is recurrent); let X_n be the latter point. Here is a global picture and a zoom (the same pictures in black and white are provided at the end of this section):

DLA   DLAzoom

Absolutely nothing is known about this model apart from the following simple result of Kesten:

    \[\limsup_{n \to +\infty} \frac{\max_{x \in A_n} \|x\|_2}{n^{2/3}} < + \infty .\]

A fascinating open problem is to show that for some \epsilon>0,

    \[\liminf_{n \to +\infty} \frac{\max_{x \in A_n} \|x\|_2}{n^{0.5+\epsilon}} = + \infty ,\]

which the above picture clearly seems to validate. Note that at the moment nobody can even prove that once rescaled by 1/\sqrt{n} the DLA will not converge to an Euclidean ball…




Ballistic DLA

Here is a model that, as far as I know, was introduced by Ronen Eldan. Instead of having one particle coming from infinity to hit the aggregate through a random walk, imagine now that the aggregate is constantly bombarded (ballistically). Precisely particles are coming from infinity in every direction (that is on every line) at a constant rate. In other words first select a direction uniformly at random, and then among the lines parallel to that direction that hit the current aggregate choose one uniformly at random. The added point is the boundary point at the intersection of this (oriented) line with the aggregate. Here is a picture where we can see some resemblance with the Eden model:


and here is a picture from further away where we can see local resemblance with DLA:


see also this zoom in black and white:


Needless to say, nothing is known about this model.


Internal Diffusion Limited Aggregation (IDLA)

Finally the IDLA model is yet another modification of DLA, where instead of starting the random walk from infinity one starts it from the origin. Perhaps surprisingly, and on the contrary to everything discussed so far, we know quite a lot about this model! Its limit shape is an actual honest Euclidean ball, see this paper by Lawler, Bramson and Griffeath. In fact we even know that the average fluctuation from the aggregate around its limit is of constant order, see this paper by Jerison, Levine and Sheffield (information about the distribution of these fluctuations is also provided!).


Writing this blog post was quite a bit of fun, and I thank Ronen Eldan, Laura Florescu, Shirshendu Ganguly, and Yuval Peres from whom I learned everything discussed here. To conclude the post here are some intriguing pictures from variants of the above model that I cooked up (unfortunately I’m not sure the models are very interesting though):



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