---
id: "20260831-0215-do-caracciolo-sportiello-2012"
title: "Do Caracciolo & Sportiello (2012) and Dickman et al. (2001) confirm the specific average-height-below-threshold figures Watkins et al. (2015) attributes to them?"
type: "capture"
status: "promoted"
origin: "batch"
promoted_to: ["30-notes/claim-caracciolo-sportiello-2012-confirms-17-8-average-height.md","30-notes/claim-dickman-2001-critical-density-corroborates-not-exactly-matches-watkins-figure.md","40-entities/entity-nicholas-w-watkins.md","40-entities/entity-grassberger.md","40-entities/entity-deepak-dhar.md","40-entities/entity-s-s-manna.md","40-entities/entity-sergio-caracciolo.md","40-entities/entity-andrea-sportiello.md","40-entities/entity-ronald-dickman.md"]
not_promoted: ["Dickman et al.'s 'threshold 1' mechanism-convention claim — folded into claim-dickman-2001-critical-density-corroborates-not-exactly-matches-watkins-figure.md as supporting detail rather than a standalone note; too thin to stand alone as atomic.","Bottom-line/verdict synthesis paragraph — a synthesis across the other two claims, not itself atomic; captured via the routed question's answered_log instead of a third note.","Further leads (Huynh et al. 2011 dimension-scaling claim, Fey-Levine-Wilson ensemble-vs-driven-system subtlety, Grassberger's own primary unlocated, Dhar 1999b Abelian-property citation unchecked) — leads, not claims; none load-bearing for a kept claim, left for a future session.","Alessandro Vespignani entity candidate — one of six Dickman et al. co-authors, no distinct role or action attributed to him beyond authorship; held back as a mention, not promoted to a hub.","Chao Tang / Kurt Wiesenfeld entity candidates — real, foundational SOC co-originators, but this capture does no independent work on them beyond naming them alongside Per Bak, who already has a hub page covering the trio; held back rather than creating near-duplicate stub pages."]
writer_model: "claude-sonnet-5"
date_created: "2026-08-31T00:00:00.000Z"
provenance: "batch run, 2026-08-31"
derived_from: []
tags: ["self-organized-criticality","complex-systems","threshold","sandpile-model","abelian-sandpile-model","manna-model","source-discipline","verification"]
sources: [{"source_url":"https://arxiv.org/pdf/1504.04991","source_sha":"0330696f7c2b6c24ab28565c937824c1c287f2ecdf03e697d6aaa34caa27414d","source_title":"25 Years of Self-Organized Criticality: Concepts and Controversies","source_author":"Nicholas W. Watkins, Gunnar Pruessner, Sandra C. Chapman, Norma B. Crosby, Henrik J. Jensen","source_date":"2015-04-20 (arXiv preprint); published Space Science Reviews 198, 3-44 (2016)","source_venue":"arXiv (cond-mat.stat-mech); Space Science Reviews","source_tier":1},{"source_url":"https://arxiv.org/pdf/1207.6074","source_sha":"0139523512ad9de73f891665a1b6a5d408f69788d8c9df6f69ff422df5640158","source_title":"Exact integration of height probabilities in the Abelian Sandpile Model","source_author":"Sergio Caracciolo, Andrea Sportiello","source_date":"2012-07-25 (v1); revised 2012-09-03 (v2)","source_venue":"arXiv (cond-mat.stat-mech)","source_tier":1},{"source_url":"https://arxiv.org/pdf/cond-mat/0101381","source_sha":"f34ddca873e2363026e36dc380d0deb3e9bd16e6cacd53bf51228a1b0e9c95eb","source_title":"Critical behavior of a one-dimensional fixed-energy stochastic sandpile","source_author":"Ronald Dickman, Mikko Alava, Miguel A. Muñoz, Jarkko Peltola, Alessandro Vespignani, Stefano Zapperi","source_date":"2001-01-24 (arXiv preprint)","source_venue":"arXiv (cond-mat.stat-mech); Physical Review E 64, 056104 (2001)","source_tier":1}]
seek_code_commit: "7d6d9ed"
---


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.

> [!note] Seek's commentary:
> The Caracciolo & Sportiello half is about as clean as sourcing gets — the
> conjecture, the confirmation, and the exact figure are all in the abstract
> of the primary. The Dickman half is the more interesting outcome precisely
> because it's not a clean yes or no: three numbers in the primary orbit the
> one number in the review, and none of them is it exactly. That's the kind
> of gap a promotion pass should carry forward rather than smooth over —
> the mechanism claim in
> [[claim-btw-near-threshold-clustering-is-1d-sandpile-coincidence-not-general-soc-signature]]
> doesn't depend on the fourth decimal place, but a note that says "0.9488(5)
> confirmed" without this caveat would be overclaiming what the primary
> actually supports.

## 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.