Pith Number

pith:6XTCRDEC

pith:2026:6XTCRDECQMTOZ53IZALIV5OF22

not attested not anchored not stored refs pending

Scaling limit and density conjecture for activated random walk on the complete graph

Harley Kaufman, Josh Meisel, Matthew Junge

The stationary number of sleeping particles in activated random walk on the complete graph has a Gumbel scaling limit for sink probabilities in the window exp(-n^{1/3}) ≪ q_n ≪ n^{-1/2}.

arxiv:2604.04747 v2 · 2026-04-06 · math.PR

Open paper page JSON Open Graph Bundle Merged state Verified badge What is a Pith Number?

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{6XTCRDECQMTOZ53IZALIV5OF22}

Prints a linked badge after your title and injects PDF metadata. Compiles on arXiv. Learn more · Embed verified badge

Record completeness

1 Bitcoin timestamp

2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

✓ Portable graph bundle live · download bundle · merged state

The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

We show that the number of sleeping particles S_n left by the stationary distribution has a Gumbel scaling limit for exp(-n^{1/3}) ≪ q_n ≪ n^{-1/2}. This implies that the stationary configuration law is not a product measure. We also prove that S_n/n converges to p if and only if q_n = e^{-o(n)}, and that, when q_n=0, the number of jumps to stabilization undergoes a phase transition at density p.

C2weakest assumption

The proofs rely on the complete-graph mean-field structure and the precise asymptotic window for q_n; if the graph were not complete or the window violated, the Gumbel limit and the iff convergence statement would not necessarily hold.

C3one line summary

Activated random walk on the complete graph with sink has Gumbel scaling for sleeping particles when exp(-n^{1/3}) ≪ q_n ≪ n^{-1/2}, density converges to p only for exponentially weak sinks, and jumps to stabilization phase-transition at density p when q_n=0.

Cited by

1 paper in Pith

Law of large numbers for activated random walk on villages

Receipt and verification

First computed	2026-05-21T01:04:25.414268Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

f5e6288c828326ecf768c8168af5c5d6963ddb5880eb15ba1d7789c1b7bf1614

Aliases

arxiv: 2604.04747 · arxiv_version: 2604.04747v2 · doi: 10.48550/arxiv.2604.04747 · pith_short_12: 6XTCRDECQMTO · pith_short_16: 6XTCRDECQMTOZ53I · pith_short_8: 6XTCRDEC

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/6XTCRDECQMTOZ53IZALIV5OF22 \
  | 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: f5e6288c828326ecf768c8168af5c5d6963ddb5880eb15ba1d7789c1b7bf1614

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "c55e42e3c889de9a594bd1643d9f0ebb52ef14cb2003af863601ac76d35d360a",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.PR",
    "submitted_at": "2026-04-06T15:13:24Z",
    "title_canon_sha256": "97cc7613beafb173aa37c4f11bb20ddd32be40c828a503f959b5836a1a7d1620"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2604.04747",
    "kind": "arxiv",
    "version": 2
  }
}