Do Caracciolo & Sportiello (2012) and Dickman et al. (2001) confirm the specific average-height-below-threshold figures Watkins et al. (2015) attributes to them?
This capture directly answers the open verification question
question-verify-caracciolo-sportiello-dickman-soc-threshold-average-heights
(raised 2026-08-30) and resolves the [unverified-quant] flag on
claim-btw-near-threshold-clustering-is-1d-sandpile-coincidence-not-general-soc-signature.
That claim-note quotes Watkins, Pruessner, Chapman, Crosby & Jensen (2015),
"25 Years of Self-Organized Criticality," §6.3, p. 18-19: "in two dimensions,
the sandpile model has been conjectured (Grassberger quoted by Dhar, 2006,
finally confirmed analytically by Caracciolo and Sportiello, 2012) to have
average height of 17/8 = 2.125, well below the threshold of 3, and the
Abelian Manna Model (Manna, 1991; Dhar, 1999b) with threshold 1 in one
dimension has average height 0.9488(5) (Dickman et al., 2001)." Both
underlying primaries were fetched via extract_pdf and read directly this
session.
Claim: Caracciolo & Sportiello (2012) exactly confirm the 17/8 average-height figure Watkins et al. attribute to them
Claim type: quantitative. Floor: Tier 1-2 required; met (Tier 1, own arXiv preprint, direct quote).
Caracciolo & Sportiello's paper states its own result in its abstract: "we confirm the predictions on the probabilities, and thus, as a corollary, the conjecture on the average height, ⟨ρ⟩ = 17/8." The paper's introduction derives this from first principles for the Abelian Sandpile Model on the square lattice, where the height variable at each site takes values zi = 0,1,2,3, and "if this height exceeds the critical value zc = 3, then the site topples" — matching the threshold of 3 that Watkins et al. cite. The paper attributes the 17/8 value's origin to a conjecture by Grassberger (reported via Dhar's review), which the paper's analytic integration of the height-probability integrals then confirms as a corollary: ⟨ρ⟩ = 7/4 + 3/(2π) − 3/π² + 3I₂/32 = 17/8. This is an exact, digit-for-digit match to the figure Watkins et al. attribute to this source.
source_quote: "we confirm the predictions on the probabilities, and thus, as a corollary, the conjecture on the average height, hρi = 17/8." (Caracciolo & Sportiello, 2012, abstract)
Claim: Dickman et al. (2001) report a critical-density figure in the same range as Watkins et al.'s "0.9488(5)," but no single number printed in the paper matches that notation digit-for-digit
Claim type: quantitative. Floor: Tier 1-2 required; met (Tier 1, own arXiv preprint, direct quote) — but the figure itself only partially corroborates, see below.
Dickman et al. study a one-dimensional stochastic ("Manna-variant") fixed-energy sandpile and locate its absorbing-to-active phase transition at a critical particle/energy density ζc. The paper reports three distinct numerical estimates, not one: "we find ζc = 0.94887(7), with the uncertainty reflecting the scatter in our numerical results for the curvature... A similar analysis of the data for ρ²a yields ζc = 0.94883(5)... We therefore adopt the estimates ζc = 0.94885(7)." None of these three values is written "0.9488(5)" in the paper itself — the adopted estimate rounds to 0.9488 or 0.9489 depending on convention, and the "(5)" uncertainty digit that Watkins et al. attach appears verbatim only on the ρ²a-derived estimate (0.94883(5)), which is not the paper's adopted final value. The figure Watkins et al. print is therefore in the correct ballpark and traceable to this paper's own numbers, but is best read as a compressed/rounded transcription rather than a literal quotation of any one reported estimate.
source_quote: "we find ζc = 0.94887(7)... A similar analysis of the data for ρ2a yields ζc = 0.94883(5)... We therefore adopt the estimates ζc = 0.94885(7)." (Dickman et al., 2001, §III.A)
Claim: Dickman et al.'s (2001) model does use the "threshold 1" toppling convention Watkins et al. describe
Claim type: technical mechanism. Floor: Tier 1-2 required; met (Tier 1, own arXiv preprint, direct quote).
Watkins et al. describe the Abelian Manna Model figure as holding for a system "with threshold 1 in one dimension." Dickman et al.'s model defines the site variable zi as "the energy (or number of walkers)... sites with zi ≥ 2 are said to be active" and topple; equivalently, a site topples once its height exceeds 1, the same "exceeds threshold value 1" convention Watkins et al. use to describe the one-dimensional BTW model earlier in their own paper (hi − hi+1 > 1). The mechanism label Watkins et al. attach to this figure is accurate to Dickman et al.'s own model definition.
source_quote: "The configuration is specified by the energy (or number of walkers) zi = 0, 1, 2, ... at each site; sites with zi ≥ 2 are said to be active." (Dickman et al., 2001, §II)
Bottom line on the core question
Partially confirmed, with one exact match and one close-but-imprecise transcription. Caracciolo & Sportiello (2012) is an exact, primary-sourced match for the 17/8 figure — no discrepancy found. Dickman et al. (2001) substantively supports the existence and rough magnitude of the ~0.9488 critical-density figure and the "threshold 1" mechanism description, but Watkins et al.'s specific "0.9488(5)" notation does not correspond to any single number the primary paper prints; it most closely resembles the ρ²a-derived estimate (0.94883(5)) rather than the paper's adopted final estimate (0.94885(7)). This is a minor precision slip in a otherwise accurate review-level citation, not a fabrication or a wrong order of magnitude — worth recording as a specific, checkable discrepancy rather than either a clean confirmation or a refutation.
Further leads
- Huynh et al. (2011), cited by Watkins et al. as reporting the Manna-model average height is "expected to drop to 1/2 with increasing dimension" — not fetched or read this session.
- Caracciolo & Sportiello's own paper flags a subtlety worth a future capture: Fey, Levine & Wilson showed a small but nonzero discrepancy (δρ/ρ ∼ 10⁻⁴) between the average height over the uniform ensemble of recurrent configurations and the density in the driven critical system with conserved mass — the 17/8 figure is for the former, not necessarily identical to the latter.
- Grassberger's original conjecture of 17/8 is reported by Watkins et al. only via Dhar's 2006 review (Dhar, D., "Theoretical studies of self-organized criticality," Physica A); the primary Grassberger source for the conjecture itself was not located or read this session.
- Watkins et al.'s Manna-model citation "Dhar, 1999b" for the model's Abelian property was not independently checked this session.
Entity candidates
- Grassberger — person — originated the 17/8 average-height conjecture for the 2D Abelian Sandpile Model that Caracciolo & Sportiello's paper exists to confirm; the foundational figure this capture's central claim is measured against, not just a co-author.
- S.S. Manna — person — originated the stochastic sandpile model (Manna, 1991) of which Dickman et al.'s 1D fixed-energy model is a variant; the foundational figure behind the second half of this capture.
- Deepak Dhar — person — cited by Watkins et al. as the source reporting Grassberger's conjecture (Dhar, 2006) and for the Abelian Manna Model reference (Dhar, 1999b); recurring bridge citation between the founding conjecture and its later confirmation.
- Sergio Caracciolo — person — co-author of the primary that analytically confirms the 17/8 figure.
- Andrea Sportiello — person — co-author of the primary that analytically confirms the 17/8 figure.
- Ronald Dickman — person — first author of the primary reporting the ~0.9488 critical-density figure for the 1D Manna-variant sandpile.
- Alessandro Vespignani — person — co-author of Dickman et al. (2001); recurring figure across multiple fixed-energy-sandpile papers cited in this thread.
- Nicholas W. Watkins — person — lead author of the 2015 Tier-1 review whose attributed figures this capture set out to verify.
- Per Bak, Chao Tang, Kurt Wiesenfeld (BTW) — people/concept — originators of the sandpile model and the self-organized-criticality concept that both underlying papers' threshold/height conventions ultimately extend from.