# Local structure in the stochastic sandpile model

Here's a question that came up at the recent AIM conference on chip-firing and generalizations.

The stochastic sandpile model, I think originally due to Manna, is a stochastic process that (in one variant) works like this. I have an infinite sequence of locations, indexed by Z.

At each stage, I drop a chip at the origin. Now every time there are two chips at any location, both chips "fire": that is, each one moves left or right with probability 1/2.

So at time 1 there's one chip at the origin.

At time 2 there are two chips at the origin; they fire, and there's a 1/4 chance of having 2 chips at -1, 1/2 chance of having chips at -1 and 1, and 1/4 chance of having 2 chips at 1. Of course, in the first and third cases, more chip-firing takes place.

And so on and so on.

If you drop a large number N of chips at the origin, and then let the system relax to stability (which it eventually does, with probability 1) you'll end up with a configuration of chips on the line in which no two chips occupy the same spot; in other words, a subset of Z.

People believe (but have not proved) that if you let this process run for a long time, there will be a large area around the origin where the configuration has density equal to the critical value, which is about .94...

But one can ask more refined questions about the configurations that are likely to arise. You are looking at an interval around the origin in which most of the locations are covered by a chip; but there are a small proportion of locations, about 6% of the total, which are empty; we call these "zeroes." Small experiments suggest that these are not independent of each other; rather, they tend to repel each other.

Question: Whether experimentally or provably, can you say anything about the joint distribution of the zeroes? Does this process resemble any of the other well-known point processes with repulsion between points? More generally, one might expect that the distribution of the restriction of the configuration to [-k, .. k] approaches a limit as N -> oo; can one say anything intelligent about what this looks like? (The conjecture on the critical value says precisely that when k=0 the configuration has a 94% chance of being [1] and a 6% chance of being [0].)

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