bibkey: riihimakismith2026avalanche
authors: Henri Riihimäki, Jason P. Smith
year: 2026
title: "Avalanche homology of digraphs via sandpile dynamics"
doi: null
url: https://arxiv.org/abs/2606.26786v1
claim: "Definitions 2.2 and 2.4 specify parallel firing of unstable nonsinks and the downward closure of all complete firing sets."
strata_touched:
  - D5/S3/StatisticalMechanics/Sandpiles/UniformClockAvalanche
license: citation-only
triage: anchor

Parallel avalanches on digraphs

Definition 2.2 fires every vertex with positive out-degree whose old number of grains is at least its out-degree. Each fires once; incoming grains become available in the next configuration. Equation (1) uses the transpose of the row-source Laplacian. Sinks absorb grains and are explicitly excluded from the firing sets. Definition 2.4 defines the avalanche complex by downward closure of the complete firing sets over every natural time.

The uniform clock construction uses these dynamics, with a clock for each entry of a surjective ordered enumeration of the maximal faces of a finite complex. Its state profiles and generated-face formula are a separate mathematical consequence; the source supplies the dynamics conventions, not that construction theorem. No homotopy or torsion conclusion is asserted here.

Verified locator

https://arxiv.org/abs/2606.26786v1