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We study a case of two competing non-trivial fixed points for a three-dimensional real $N$-component field: the O(N)-invariant fixed point vs.~the cubic-invariant fixed point. We compute the critical value $N_{c}$ of the cubic $\\phi^{4}$-perturbation at the O(N)-fixed point. The O(N) fixed point is stable under a cubic $\\phi^{4}$-perturbation below $N_{c}$, above $N_{c}$ it is unstable. The critical value comes out as $2.219435<N_{c}< 2.219436$ in the ultralocal approximation. We also compute the c"},"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":"cond-mat/9805193","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"cond-mat.stat-mech","submitted_at":"1998-05-15T16:47:52Z","cross_cats_sorted":[],"title_canon_sha256":"b50bef31ecd6ad1596215a013bb6cce7ca7dfc39014efc01fd9a4504ebc9c4a6","abstract_canon_sha256":"f58a4eb670ce49df2c5c4426c9291ba62fb8550b1aab31e76dea056c07590a03"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:39:30.507295Z","signature_b64":"zSaQbk85fuby0VQ1xnF1M2aBN8ztJP5IIw9HloSXiRBpXsXBgE+s3Jp4AEMj7OesnF7RGijZ0zDMXn4UMcRUAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"efe6b3bf75ec530f0125b47fe1cec7461419c7ccfc822a9b91b8384b02700cd1","last_reissued_at":"2026-05-18T01:39:30.506557Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:39:30.506557Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Stability of the O(N)-Invariant and the Cubic-Invariant 3-Dimensional $N$-Component Renormalization Group Fixed Points in the Hierarchical Approximation","license":"","headline":"","cross_cats":[],"primary_cat":"cond-mat.stat-mech","authors_text":"C. Wieczerkowski (University of M\\\"unster, Germany), K. Pinn, M. Rehwald","submitted_at":"1998-05-15T16:47:52Z","abstract_excerpt":"We compute renormalization group fixed points and their spectrum in an ultralocal approximation. We study a case of two competing non-trivial fixed points for a three-dimensional real $N$-component field: the O(N)-invariant fixed point vs.~the cubic-invariant fixed point. We compute the critical value $N_{c}$ of the cubic $\\phi^{4}$-perturbation at the O(N)-fixed point. The O(N) fixed point is stable under a cubic $\\phi^{4}$-perturbation below $N_{c}$, above $N_{c}$ it is unstable. The critical value comes out as $2.219435<N_{c}< 2.219436$ in the ultralocal approximation. 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