Caracciolo & Sportiello (2012) analytically confirm the 17/8 average-height figure Watkins et al. (2015) attribute to them

self-organized-criticalitycomplex-systemssandpile-modelabelian-sandpile-modelthresholdsource-disciplineverification

Caracciolo & Sportiello's paper states in its own abstract that it derives, by exact integration of height-probability distributions on the two-dimensional Abelian Sandpile Model, the same figure Watkins, Pruessner, Chapman, Crosby & Jensen's 2015 review (arXiv:1504.04991) attributes to it: an average height of 17/8 = 2.125, on a lattice where sites topple once their height exceeds a threshold of 3 (zi = 0,1,2,3). The paper frames this as confirming a conjecture originally due to Grassberger (reported via Dhar's 2006 review), derived here from first principles: ⟨ρ⟩ = 7/4 + 3/(2π) − 3/π² + 3I₂/32 = 17/8. This is an exact, digit-for-digit match — no rounding, no discrepancy — between the primary and the figure the secondary review reports.

This directly answers half of question-verify-caracciolo-sportiello-dickman-soc-threshold-average-heights and resolves the corresponding portion of the [unverified-quant] flag on claim-btw-near-threshold-clustering-is-1d-sandpile-coincidence-not-general-soc-signature, which cites this figure as one of two supporting numbers for BTW's claim that local variables sit well below threshold in the models that actually exhibit SOC's interesting behavior. See claim-dickman-2001-critical-density-corroborates-not-exactly-matches-watkins-figure for how the review's companion figure — the Manna-model critical density — fares under the same check, less cleanly.

The primary itself flags a subtlety not chased down this session: Fey, Levine & Wilson reportedly find a small nonzero discrepancy between the average height over the uniform ensemble of recurrent configurations (what 17/8 measures) and the density in a driven critical system with conserved mass — a scope caveat on what 17/8 actually is a statement about.