{"paper":{"title":"Perfect simulation for interacting Hawkes processes with reset-induced variable length memory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"If the sure-event rate exceeds the candidate-event rate in interacting Hawkes processes, their clans of ancestors are finite almost surely.","cross_cats":[],"primary_cat":"math.PR","authors_text":"Branda P.I. Gon\\c{c}alves, Lucien Mauffret","submitted_at":"2026-05-13T13:37:00Z","abstract_excerpt":"We study a class of interacting nonlinear Hawkes point processes on the integer lattice in which each component is reset after its own jumps. The intensity of a component depends on the post-reset activity of its nearest neighbours, which produces a variable-length memory structure.\n  We develop a graphical construction based on a dominating Poisson environment and introduce the clan of ancestors of a space-time point. The clan is the finite or infinite backward exploration of all events whose acceptance decisions may influence the target value. Our main result is a constructive subcriticality"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"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.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"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.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"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.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"If the sure-event rate exceeds the candidate-event rate in interacting Hawkes processes, their clans of ancestors are finite almost surely.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"045cdc4e0b84f20167d7986cf4fa74bc1f064b02914d1241d04ab6133103f21b"},"source":{"id":"2605.13519","kind":"arxiv","version":1},"verdict":{"id":"7a77455a-cad2-4e41-85c5-e9ff210c24ff","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T18:09:48.034797Z","strongest_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.","one_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.","pipeline_version":"pith-pipeline@v0.9.0","weakest_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.","pith_extraction_headline":"If the sure-event rate exceeds the candidate-event rate in interacting Hawkes processes, their clans of ancestors are finite almost surely."},"references":{"count":12,"sample":[{"doi":"","year":1972,"title":"Athreya, K.B. and Ney, P.E. (1972).Branching Processes. Springer, Berlin","work_id":"60cea247-8248-4069-902c-17cde4d260e7","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1996,"title":"P. Brémaud and L. Massoulié. Stability of nonlinear Hawkes processes.Ann. Probab.24(1996), 1563–1588","work_id":"a717ae22-4b5a-4dd5-9d84-04a3ea732ccc","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2002,"title":"F. Comets, R. Fernandez and P.A. Ferrari. Processes with long memory: regenerative construction and perfect simulation.Ann. Appl. Probab.12(2002), 921–943","work_id":"32a5dceb-f1f9-4a08-b6d7-2b54fcdd749a","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2016,"title":"S. Delattre, N. Fournier and M. Hoffmann. Hawkes processes on large networks.Ann. Appl. Probab.26(2016), 216–261","work_id":"5d951fba-0405-49d5-a6ba-4d34bdc6031f","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2018,"title":"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","work_id":"32c18e19-aa2e-4204-8efa-074437579138","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":12,"snapshot_sha256":"d9da8c65be0d7c0930ad72410996b648906627744994e32389e43a3eeb782122","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"d048cb4c38617016d414d16f01a91bd9bd1701242eaf6413b7bfd9a2820f6efc"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}