Random Interlacements

In this post, I want to talk about another recently-introduced model that’s generating a lot of interest in probability theory, Sznitman’s model of random interlacements. We also want to see, at least heuristically, how this relates to more familiar models.

We fix our attention on a lattice, which we assume to be \mathbb Z ^d. We are interested in the union of an infinite collection of simple random walks on the lattice. The most sensible thing to consider is not a collection of random walks from at a random set of starting points, but rather a family of trajectories, that is a doubly-infinite random walk defined on times (-\infty,\infty). We will want this family to have some obvious properties, such as translation invariance, in order to make analysis possible and ideally obtain some 0-1 laws. The natural thing to do is then to choose the trajectories through a Poisson Point Process. The tricky part will be finding an intensity measure that has all the properties we want, and gives trajectories that genuinely do look like SRWs, and, most importantly, have a union that is neither too sparse nor too dense. For example, it wouldn’t be very interesting if with high probability every point appeared in the union…

For reasons we will mention shortly, we are interested in the complement of the union of the trajectories. We call this the vacant set. We will find an intensity which we can freely scale by some parameter u\in\mathbb{R}^+, which will give us a threshold for the complement to contain an infinite component. This is in the same sense as the phase transition for Bernoulli percolation. That is, there is a critical value u^* say, such that for u<u^* the vacant set contains an infinite component (or percolates) almost surely, and almost surely it does not when u>u^*. A later result of Teixeira shows that, as in percolation, this infinite component is unique.

Let us first recall why it is not interesting to consider this process for d=1 or 2. On \mathbb{Z}, with high probability a single SRW hits every integer point trivially, since it visits arbitrarily large and arbitrarily small integers. For d=2, the SRW is recurrent, and so consists of a countably infinite sequence of excursions from (0,0). Note that the probability that an excursion from 0 hits some point (x,y) is non-zero, as it is at least 2^{-2(|x|+|y|)} for example. Therefore, with high probability the SRW hits (x,y), and so whp it hits every point.

Therefore it is only for d=>3 that we start seeing interesting effects. It is worth mentioning at this point some of the problems that motivated considering this model. First is the disconnection time of a discrete cylinder by a simple walk. For example, Sznitman considers the random walk on \mathbb{Z}\times (\mathbb{Z}/N\mathbb{Z})^d. Obviously, it is more interesting to consider how long it takes a (1-dimensional in the natural sense) path to disconnect a d=>3 dimensional set than a 2-dimensional one, as the latter is given just by the first time the path self-intersects.

More generally, we might be interested in random walks up to some time an order of magnitude smaller than the cover time. Recall the cover time is the time to hit each point of the set. For example, for the random walk on the d-dimensional torus (\mathbb{Z}/N\mathbb{Z})^d the cover time (as discussed in Markov Chains and Mixing Times posts) is N^d \log N, but the log N represents in some sense only the ‘final few’ vertices. So we should ask what the set of unhit vertices looks like at time N^d. And it turns out that for large N, the structure of this vacant set is related to the vacant set in the random interlacement model, in a local sense.

Anyway, the main question to ask is: what should the intensity measure be?

We patch it together locally. Start with the observation that transience of the random walk means almost surely a trajectory spends only finitely many steps in a fixed finite set K. So we index all the trajectories which hit K by the first time they hit K. Given that a trajectory hits K, it is clear what the conditional distribution of this hitting point should be. Recall that SRW on Z^d is reversible, so we consider the SRW backwards from this hitting then. Then the probability that the hitting point is x (on the boundary of K) is proportional to the probability that a SRW started from x goes to infinity without hitting K again. So once we’ve settled on the distribution of the hitting point x, it is clear how to construct all the trajectories through K. We pick x on the boundary of K according to this distribution, and take the union of an SRW starting from x conditioned not to hit K again, and an SRW starting from x with no conditioning. These correspond to the trajectory before and after the hitting time, respectively.

In fact, it turns out that this is enough. Suppose we demand that the probability that the hitting point is x is equal to the probability that a SRW started from x goes to infinity without hitting K again (rather than merely proportional to). Sznitman proves that there is a unique measure on the set of trajectories that restricts to this measure for every choice of K. Furthermore, the Poisson Point Process with the globally-defined intensity, unsurprisingly restricts to a PPP with the intensity specific to K.

We have not so far said anything about trajectories which miss this set K. Note that under any sensible intensity with the translation-invariance property, the intensity measure of the trajectories which hit K must be positive, since we can cover \mathbb{Z}^d with countably many copies of K. So the number of trajectories hitting K is a Poisson random variable.

Recall how we defined the probability that the hitting point of K was some point x on the boundary. The sum of these probability is called the capacity of K. It follows that this is the parameter of the Poisson random variable. Ie, the probability that no trajectory passes through K is:

\exp(-u\mathrm{cap}(K)),

recalling that u is the free parameter in the intensity. This is the most convenient framework through which to start analysing the probability that there is an infinite connected set which is hit by no trajectory.

We conclude by summarising Sznitman’s Remark 1.2, explaining why it is preferable to work with the space of trajectories rather than the space of paths. Note that if we are working with paths, and we want translation invariance, then this restricts to translation invariance of the distribution of starting points as well, so it is in fact a stronger condition. Note then that either the intensity of starting at 0 is zero, in which case there are no trajectories at all, or it is positive, in which case the set of starting points looks like Bernoulli site percolation.

However, the results about capacity would still hold if there were a measure that restricted satisfactorily. And so the capacity of K would still be the measure of paths hitting K, which would be at least the probability that the path was started in K. But by translation invariance, this grows linearly with |K|. But capacity grows at most as fast as the size of the set of boundary points of K, which will be an order of magnitude smaller when K is, for example, a large ball.

REFERENCES

This was mainly based on

Sznitman – Vacant Set of Random Interlacements and Percolation (0704.2560)

Also

Sznitman – Random Walks on Discrete Cylinders and Random Interlacements (0805.4516)

Teixeira – On the Uniqueness of the Infinite Cluster of the Vacant Set of Random Interlacements (0805.4106)

and some useful slides by the same author (teixeira.pdf)

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