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Numerical simulations were carried out on a two dimensional lattice $L\\times L$ in which two constant thresholds $E_{c}^{I}=4$ and $E_{c}^{II}>E_{c}^{I}$ were randomly distributed. A density of sites $c$ with the threshold $E_{c}^{II}$ and threshold $E_{c}^{II}$ are parameters of the model. I have determined autocorrelations of avalanche size waves, Hurst exponents, avalanche structures and avalanche siz"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"0908.0318","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.stat-mech","submitted_at":"2009-08-03T19:24:55Z","cross_cats_sorted":[],"title_canon_sha256":"22559756ebbc95d2d93f44e21f911788c733aa872aa818223998f6844e8393e1","abstract_canon_sha256":"178fce8075a9606f5439268c687c4f7960dcec6694b7d957f8d107c928878b8b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:12:45.024492Z","signature_b64":"dA+Q6VMV2szDInnjYdFMA/RIUnDhwy2MzA4t3P9++xJ3xcWY7oHUmCZgX7eWtO07m42SC+xb1T5O3vRDx4lXDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"748f204234e5fb504d0e337fc5a06b29ca3556244b4acc0291bf9865edc601ce","last_reissued_at":"2026-05-18T02:12:45.023915Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:12:45.023915Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Abelian deterministic self organized criticality model: Complex dynamics of avalanche waves","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cond-mat.stat-mech","authors_text":"Jozef Cernak","submitted_at":"2009-08-03T19:24:55Z","abstract_excerpt":"The aim of this study is to investigate a wave dynamics and size scaling of avalanches which were created by the mathematical model {[}J. \\v{C}ern\\'ak Phys. 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