Fixed point forms of the parallel symmetric sandpile model

This paper presents a generalization of the sandpile model, called the parallel symmetric sandpile model, which inherits the rules of the symmetric sandpile model and implements them in parallel. In this new model, at each step the collapsing of the collapsible columns happens at the same time and one collapsible column is able to collapse on the left or on the right but not both. We prove that the set of forms of fixed points of the symmetric sandpile model is the same as the one of that model using parallel update scheme by constructing explicitly the way (in the parallel update scheme) to reach the form of an arbitrary fixed point of the sequential model.

💡 Research Summary

The paper introduces a new variant of sandpile dynamics called the Parallel Symmetric Sandpile Model (PSSPM). This model inherits the rules of the Symmetric Sandpile Model (SSPM), in which grains may topple to either the left or the right, but applies them in a parallel fashion: at each discrete time step every collapsible column topples simultaneously, and if a column is collapsible on both sides it must choose exactly one direction. The authors first review the classical Sandpile Model (SPM), its parallel version (PSPM), and the symmetric version (SSPM). SPM is deterministic, right‑only, and converges to a unique fixed point in Θ(n³/2) steps. PSPM runs all possible right‑collapses in parallel, reducing the convergence time to Θ(n) while preserving the same unique fixed point. SSPM is non‑deterministic because columns may collapse left or right; consequently it admits multiple fixed points. Prior work showed that the number of distinct fixed‑point forms (ignoring translation) for SSPM with n grains is ⌊√n⌋, and each fixed point has height either ⌊√n⌋ or ⌊√n⌋−1.

The main contribution of the present work is Theorem 5, which states that the set of fixed‑point forms reachable by PSSPM is exactly the same as that of SSPM. In other words, although PSSPM has fewer actual fixed points (because the parallel rule eliminates some positional variants), it can still produce every shape that SSPM can, up to translation. Moreover, the authors provide an explicit construction that, given any SSPM fixed‑point form, yields a sequence of PSSPM transitions from the initial single‑column configuration (n) to a configuration of that form. The construction is essentially optimal in length, i.e., it is “nearly shortest”.

The proof is organized around three algorithmic procedures:

1. Pseudo‑Alternating Procedure – Starting from (n), the system is driven to an intermediate configuration Q that mirrors the weight imbalance d = |w(P>0) – w(P<0)| of the target fixed point P. This procedure repeatedly applies the parallel rule while adding a grain to the leftmost column on odd steps (the “Atom” sub‑procedure). The result is a symmetric “mountain” shape whose central column has height n−d²/2 and whose left/right wings have lengths d−1, d−2, …, 1.

2. Alternating Procedure – From Q the system alternates left‑ and right‑collapses of the central column. On odd steps the central column topples to the right, on even steps to the left. This continues until no further alternating move is possible, yielding a configuration R whose height is exactly the height h of the desired fixed point. Lemma 7 shows that R is uniquely determined by d and h.

3. Deterministic Procedure – In R every column is collapsible on at most one side, so the remaining evolution is deterministic. By repeatedly applying the parallel rule (which now coincides with the sequential rule) the system converges to the target fixed point P. Lemma 8 guarantees that this phase never re‑introduces a column that could collapse on both sides.

For symmetric fixed points (where the left and right parts are exact reverses of each other) the construction simplifies: the Alternating Procedure alone, starting from (n), already yields the desired shape, as shown in Corollary 3. For non‑symmetric fixed points the three‑phase scheme is required, but the total number of steps remains O(n) and is provably close to the theoretical minimum.

The paper also discusses the relationship between the number of fixed points and fixed‑point forms. While PSSPM has strictly fewer fixed points than SSPM (because some positional variants are eliminated by the parallel rule), the number of distinct forms remains ⌊√n⌋, matching the known result for SSPM (Theorem 4). This demonstrates that parallel updating does not reduce the combinatorial richness of the model’s asymptotic shapes, even though it dramatically speeds up convergence.

In conclusion, the authors have shown that parallelizing the symmetric sandpile dynamics preserves the full spectrum of attainable fixed‑point shapes while offering a linear‑time convergence to a fixed point. The constructive proof provides explicit, near‑optimal sequences of moves, which could be useful for algorithmic implementations in cellular automata, chip‑firing games, or physical simulations of self‑organized criticality. The paper suggests several avenues for future work, such as extending the analysis to multi‑column initial configurations, stochastic choices for the direction of ambiguous collapses, or higher‑dimensional lattice versions of the model.