Pith Number

pith:576XC73F

pith:2026:576XC73FFTOSK4EXQXEFQNHCLL

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The critical activation density in graph bootstrap percolation

Brett Kolesnik, Maksim Zhukovskii, Pawe{\l} Pra{\l}at, Rajko Nenadov, Tam\'as Makai, Xavier P\'erez-Gim\'enez

For every graph H the critical H-percolation threshold p_c(n,H) is located in terms of the limiting density ρ(H) of the graphs that activate an edge most efficiently.

arxiv:2605.15066 v1 · 2026-05-14 · math.PR · math.CO

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Claims

C1strongest claim

for every graph H, we locate the critical H-percolation threshold p_c(n,H)

C2weakest assumption

That the critical limiting density ρ(H) of graphs that most efficiently activate a given edge is well-defined and finite for every H, and that the threshold is determined by this quantity.

C3one line summary

For any fixed graph H the critical percolation threshold p_c(n,H) equals the value determined by the minimal activation density ρ(H) of graphs that efficiently activate an edge.

References

30 extracted · 30 resolved · 0 Pith anchors

[1] M. Aizenman and J. L. Lebowitz,Metastability effects in bootstrap percolation, J. Phys. A21(1988), no. 19, 3801–3813 1988

[2] N. Alon,An extremal problem for sets with applications to graph theory, J. Combin. Theory Ser. A40(1985), no. 1, 82–89 1985

[3] N. Alon and J. H. Spencer,The probabilistic method, fourth ed., Wiley Series in Discrete Mathematics and Optimization, John Wiley & Sons, Inc., Hoboken, NJ, 2016 2016

[4] O. Angel and B. Kolesnik,Sharp thresholds for contagious sets in random graphs, Ann. Appl. Probab.28(2018), no. 2, 1052–1098 2018

[5] ,Large deviations for subcritical bootstrap percolation on the Erd˝ os–Rényi graph, J. Stat. Phys.185(2021), no. 2, Paper No. 8, 16 2021

Receipt and verification

First computed	2026-05-17T23:38:54.229160Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

effd717f652cdd25709785c85834e25ad6da6f7d926ee6c1cf4cc2ab25bea2d4

Aliases

arxiv: 2605.15066 · arxiv_version: 2605.15066v1 · doi: 10.48550/arxiv.2605.15066 · pith_short_12: 576XC73FFTOS · pith_short_16: 576XC73FFTOSK4EX · pith_short_8: 576XC73F

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/576XC73FFTOSK4EXQXEFQNHCLL \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: effd717f652cdd25709785c85834e25ad6da6f7d926ee6c1cf4cc2ab25bea2d4

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "02af2e2dc014bd743712f2bbe6b1cd8d60f293039b69f508eaf8a90b0a61084b",
    "cross_cats_sorted": [
      "math.CO"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.PR",
    "submitted_at": "2026-05-14T16:56:35Z",
    "title_canon_sha256": "372081e84101975713fd6fd29401cf8a59eb10c946fbf4918c1621f8733569d7"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.15066",
    "kind": "arxiv",
    "version": 1
  }
}