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Suppose we infect any healthy vertex $v\\in [L]^d$ already having $r$ infected neighbours, and that infected sites remain infected forever. In this paper we determine the $(d-1)$-times iterated logarithm of the critical length for percolation up to a constant factor, for all $d$-tuples $(a_1,\\dots ,a_d)$ and all $r\\"},"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":"2201.09029","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2022-01-22T11:59:22Z","cross_cats_sorted":[],"title_canon_sha256":"795dbb66f58969fe40b6e6cbd82015516cbef255186e36939bf273553dc76371","abstract_canon_sha256":"c5eee4062b3daeec1f65fa9276f70d2d3f1762362f306ecda3fcb903ade3f71a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:50:39.515924Z","signature_b64":"5xFWigWZzjgmoPgES+PrbsM5/DEBexic6tzQSuYW/128tIWhpnpimOpLb8WLNzEKS7tMSFuFshx9TcuZmRYwCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f679c0d7cb8e13fde0128c594444aeb6e50c84b704afe3968a7387eb93c68f2f","last_reissued_at":"2026-07-05T03:50:39.515566Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:50:39.515566Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The $d$-dimensional bootstrap percolation models with threshold at least double exponential","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Daniel Blanquicett","submitted_at":"2022-01-22T11:59:22Z","abstract_excerpt":"Consider a $p$-random subset $A$ of initially infected vertices in the discrete cube $[L]^d$, and assume that the neighbourhood of each vertex consists of the $a_i$ nearest neighbours in the $\\pm e_i$-directions for each $i \\in \\{1,2,\\dots, d\\}$, where $a_1\\le a_2\\le \\dots \\le a_d$. 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