The famous **chip firing game** (which is closely related to sandpile models) goes like this:

Place chips at the vertices of a graph. REPEATEDLY: If a vertex $v$ of degree $d_{v}$ has at least $d_{v}$ chips, then it "fires" a chip along each of its incident edges. If there is vertex which can fire, the game is over.

Remarkably, whether a game terminates only depends on the graph and the initial configuration of chips (and **not** on the vertices chose to fire during the game). Even more remarkably, the length of the game (the number of fired vertices) does not depend on the actual play but rather only on the graph and the initial configuration.

Tardos and Bjorner-Lovasz-Shor have given well-known bounds for the length of the game.

On the other hand, there is a vast literature which studies the "critical group" (aka "sandpile group") of a graph, which is strictly speaking for a slightly different but related game (the so-called **dollar game**).

QUESTION: I am wondering if there is some connection between the duration of the chip firing game and the order of the critical group.

This is based on the vague intuition that the larger the group, the more opportunities the game has to terminate and so it can be expected to end rather sooner than later. But I may be completely off the mark here (for instance, I am mixing both games in my reasoning...).

Any thoughts on this highly appreciated.

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