Pith Number

pith:YEXPPMZQ

pith:2026:YEXPPMZQDQRYP2NU3GKDMWKQED

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Perfect simulation for interacting Hawkes processes with reset-induced variable length memory

Branda P.I. Gon\c{c}alves, Lucien Mauffret

If the sure-event rate exceeds the candidate-event rate in interacting Hawkes processes, their clans of ancestors are finite almost surely.

arxiv:2605.13519 v1 · 2026-05-13 · math.PR

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Record completeness

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2 Internet Archive

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4 Citations open

5 Replications open

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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

Our main result is a constructive subcriticality criterion: if the sure-event rate exceeds the candidate-event rate, equivalently if β_*/(β^*−β_*)>1, then the clan is almost surely finite. The finiteness of the clan yields a measurable local construction of the stationary regime.

C2weakest assumption

The graphical construction admits a dominating Poisson environment whose intensity bounds the nonlinear Hawkes intensities uniformly enough for the associated branching process to dominate the clan size; this domination must hold for the chosen nearest-neighbor interaction and reset rule.

C3one line summary

A subcriticality criterion β_*/(β^* - β_*) > 1 guarantees almost-sure finiteness of the clan of ancestors, enabling an exact backward-forward perfect simulation algorithm for the stationary regime of these variable-memory Hawkes processes.

References

12 extracted · 12 resolved · 0 Pith anchors

[1] Athreya, K.B. and Ney, P.E. (1972).Branching Processes. Springer, Berlin 1972

[2] P. Brémaud and L. Massoulié. Stability of nonlinear Hawkes processes.Ann. Probab.24(1996), 1563–1588 1996

[3] F. Comets, R. Fernandez and P.A. Ferrari. Processes with long memory: regenerative construction and perfect simulation.Ann. Appl. Probab.12(2002), 921–943 2002

[4] S. Delattre, N. Fournier and M. Hoffmann. Hawkes processes on large networks.Ann. Appl. Probab.26(2016), 216–261 2016

[5] P. A. Ferrari, A. Galves, I. Grigorescu and E. Löcherbach. Phase transition for infinite systems of spiking neurons.J. Stat. Phys. 172(2018), 1564–1575 2018

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-18T02:44:24.402269Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

c12ef7b3301c2387e9b4d99436595020f0a800d1d34a237df48e04ab564de44b

Aliases

arxiv: 2605.13519 · arxiv_version: 2605.13519v1 · doi: 10.48550/arxiv.2605.13519 · pith_short_12: YEXPPMZQDQRY · pith_short_16: YEXPPMZQDQRYP2NU · pith_short_8: YEXPPMZQ

Agent API

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/YEXPPMZQDQRYP2NU3GKDMWKQED \
  | 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: c12ef7b3301c2387e9b4d99436595020f0a800d1d34a237df48e04ab564de44b

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "03a7652590d9aaf7ac31fe227ae38c36b5f136e591ce2c39c195b2374f605de6",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.PR",
    "submitted_at": "2026-05-13T13:37:00Z",
    "title_canon_sha256": "f90df7077dde8c825e85809f46d4d6852a5ce1ba3833ccdd343f0788e5dca348"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.13519",
    "kind": "arxiv",
    "version": 1
  }
}