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 $p(n)=\lim_{k\to \infty} a(k,n)/b(k)$.

Question 1. Does the limit 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.