## Abelian sandpile models

This question is about a popular probabilistic model on graphs studied in physics, mostly, for the standard lattice in ${\mathbb R}^n$ but also on other graphs (this model is of the same spirit as percolation, Ising model, etc.).

Consider an infinite $d$-regular transitive rooted graph $\Gamma=(V,E)$ and represent it as an increasing union of finite rooted graphs $\Gamma_k$. For each $k$ consider the standard Abelian sandpile model on $\Gamma_k$ (with a sink vertex added as usual). Let $b(k)$ be the number of recurrent configurations on $\Gamma_k$, $a(k,n)$ be the number of recurrent configurations such that adding a piece of sand at the root causes an avalanche that involves toppling of $n$ vertices in $\Gamma_k$ (all terminology can be found in the paper above). For every $n$ let $p(n,k)= a(k,n)/b(k)$.

Question 1. Does the limit $p(n)=\lim_{k\to \infty} p(n,k)$ exist?

Question 2. Does the limit (limsup or liminf) depend essentially on the sequence $\Gamma_k$? For example, is it true that any two functions $p(n)$ of $\Gamma$ corresponding to different $(\Gamma_k)$ and $(\Gamma_k')$ have the same growth?

I suspect that physicists who study these models would say that the answer to both questions is obviously "yes". But I have not seen a formal proof. Still the literature is large so I might have missed something.

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How can the answer to "Why does the limit exist?" be "(obviously) yes"? Did you mean to ask "Does the limit exist?"? – Noam D. Elkies Sep 27 2011 at 1:26
@Noam: Of course you are right. I'll fix the question. – Mark Sapir Sep 27 2011 at 1:42
There are situations when the sandpile groups of $\Gamma_k$'s form an inverse system (e.g. when $\Gamma_k$ are binary de Bruijn digraphs, for which Lionel Levine actually computed these groups (math.cornell.edu/~levine/directed-line.pdf), the homomorphisms are just taking $2^{i-j}$ powers ). IMHO the corresponding profinite group ought to be connected with $\Gamma$ – perhaps it's the right way to define its sandpile group? Perhaps this might help to understand the question asked :-) – Dima Pasechnik Nov 18 2011 at 8:18
@Dima: Do these groups always form an inverse system? In that case a weaker question would indeed be whether the inverse limit does not depend on the sequence of subgraphs. Anyway, the question was related to my student's paper front.math.ucdavis.edu/1110.6263. I just asked him to define the sandpile model not for an infinite graph but for a sequence of finite graphs converging to an infinite graph. This way there is not ambiguity. – Mark Sapir Nov 18 2011 at 11:52
@Mark: these de Bruijn digraphs, they do form an inverse system. In the general setting of L.Levine's paper, probably, too, it is true... I guess I know what the inverse limit in the de Bruijn digraphs case looks like, but have no clue how to confirm this. Neither I know if this limit can be realized as the sandpile group; I should look at the article of your student. – Dima Pasechnik Nov 18 2011 at 14:36