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pith:2026:RBDBGBKRPKCSGW52ODXGK5QW64

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Percolation transition of strongly connected clusters in finite dimensions and on complete graphs

Ming Li, Qi Wang

Critical strongly connected clusters in directed percolation remain fractal across all dimensions with a change at six.

arxiv:2605.16987 v1 · 2026-05-16 · cond-mat.stat-mech

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Claims

C1strongest claim

These results show that critical SCCs remain well-defined fractal objects across dimensions, while their approach to the mean-field limit involves nontrivial changes in cluster statistics.

C2weakest assumption

The simulations correctly locate the percolation transition and extract fractal dimensions and size-distribution exponents without dominant finite-size or sampling artifacts that would alter the reported scaling relations.

C3one line summary

Simulations show critical strongly connected clusters remain fractal objects with dimension-dependent scaling: hyperscaling below d=6, mean-field above, and double power-law scaling on complete graphs.

References

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[1] The terms (p−p c)kLk/ν describe the scaling behavior approaching criticality, whileL −ωk accounts for finite-size corrections atp c

[2] In all finite dimensions,C 1 exhibits a clear power-law scaling, C1 ∼L dSCC ∼V dSCC/d,(9) indicating a continuous percolation transition of SCCs, with fractal dimensiond SCC 2048

[3] The mean- field valued SCC/d= 1/3 [40, 41] givesd SCC = 2 ind= 6

[4] The size distribution of SCCs Besides the largest SCC, we now examine the size dis- tribution of SCCs at criticality. In standard percolation 7 101 102 103 104 105 106 100 10−6 10−12 n(s, L) » s−2.108 2048

[5] Fractal dimensions of ICs and OCs We further find that both ICs and OCs exhibit fractal geometry. Applying the same finite-size scaling analysis, their fractal dimensions are consistent with that of o

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Receipt and verification

First computed	2026-05-20T00:03:34.571787Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

88461305517a85235bba70ee657616f73e23fb5771b5cedcff0620ea61331b0c

Aliases

arxiv: 2605.16987 · arxiv_version: 2605.16987v1 · doi: 10.48550/arxiv.2605.16987 · pith_short_12: RBDBGBKRPKCS · pith_short_16: RBDBGBKRPKCSGW52 · pith_short_8: RBDBGBKR

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curl -sH 'Accept: application/ld+json' https://pith.science/pith/RBDBGBKRPKCSGW52ODXGK5QW64 \
  | 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: 88461305517a85235bba70ee657616f73e23fb5771b5cedcff0620ea61331b0c

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "00c1a78fa01d72f88565f58334740eab9f6118f89a114fb5685e3077b5f27882",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "cond-mat.stat-mech",
    "submitted_at": "2026-05-16T13:14:40Z",
    "title_canon_sha256": "3d8f2359a0e95d76ac1f9528aca037b980c1c04172afc0a7696abd4cbbf427a2"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.16987",
    "kind": "arxiv",
    "version": 1
  }
}