{"id":"https://openalex.org/W7208838836","doi":"https://doi.org/10.48550/arxiv.2609.04006","title":"Quenched complexity of marginal states in the Sherrington--Kirkpatrick spin glass","display_name":"Quenched complexity of marginal states in the Sherrington--Kirkpatrick spin glass","publication_year":2026,"publication_date":"2026-09-03","ids":{"openalex":"https://openalex.org/W7208838836","doi":"https://doi.org/10.48550/arxiv.2609.04006"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.04006","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.04006","pdf_url":"https://arxiv.org/pdf/2609.04006","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.04006","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5150085206","display_name":"Tiziana De Chirico","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I4210091015","display_name":"Istituto di Nanotecnologia","ror":"https://ror.org/00bc51d88","country_code":"IT","type":"facility","lineage":["https://openalex.org/I4210091015","https://openalex.org/I4210155236"]},{"id":"https://openalex.org/I861853513","display_name":"Sapienza University of Rome","ror":"https://ror.org/02be6w209","country_code":"IT","type":"education","lineage":["https://openalex.org/I861853513"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"De Chirico, Tiziana","raw_affiliation_strings":["Dipartimento di Fisica , Sapienza Universit\u00e0 di Roma , Piazzale Aldo Moro 5 , I-00185 Rome , Italy","Institute of Nanotechnology , CNR-NANOTEC , via Monteroni , Lecce 73100 , Italy"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Dipartimento di Fisica , Sapienza Universit\u00e0 di Roma , Piazzale Aldo Moro 5 , I-00185 Rome , Italy","institution_ids":["https://openalex.org/I861853513"]},{"raw_affiliation_string":"Institute of Nanotechnology , CNR-NANOTEC , via Monteroni , Lecce 73100 , Italy","institution_ids":["https://openalex.org/I4210091015"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5049776530","display_name":"Luca Leuzzi","orcid":"https://orcid.org/0000-0002-5971-7191","observed_orcids":["https://orcid.org/0000-0002-5971-7191"]},"institutions":[{"id":"https://openalex.org/I4210091015","display_name":"Istituto di Nanotecnologia","ror":"https://ror.org/00bc51d88","country_code":"IT","type":"facility","lineage":["https://openalex.org/I4210091015","https://openalex.org/I4210155236"]},{"id":"https://openalex.org/I4210155236","display_name":"National Research Council","ror":"https://ror.org/04zaypm56","country_code":"IT","type":"nonprofit","lineage":["https://openalex.org/I4210155236"]},{"id":"https://openalex.org/I861853513","display_name":"Sapienza University of Rome","ror":"https://ror.org/02be6w209","country_code":"IT","type":"education","lineage":["https://openalex.org/I861853513"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"Leuzzi, Luca","raw_affiliation_strings":["Dipartimento di Fisica , Sapienza Universit\u00e0 di Roma , Piazzale Aldo Moro 5 , I-00185 Rome , Italy","Institute of Nanotechnology , CNR-NANOTEC , Soft and Living Matter Laboratory , Piazzale Aldo Moro 5 , I-00185 Rome , Italy"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Dipartimento di Fisica , Sapienza Universit\u00e0 di Roma , Piazzale Aldo Moro 5 , I-00185 Rome , Italy","institution_ids":["https://openalex.org/I861853513"]},{"raw_affiliation_string":"Institute of Nanotechnology , CNR-NANOTEC , Soft and Living Matter Laboratory , Piazzale Aldo Moro 5 , I-00185 Rome , Italy","institution_ids":["https://openalex.org/I4210091015","https://openalex.org/I4210155236"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10591","display_name":"Theoretical and Computational Physics","score":0.9663000106811523,"subfield":{"id":"https://openalex.org/subfields/3104","display_name":"Condensed Matter Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10591","display_name":"Theoretical and Computational Physics","score":0.9663000106811523,"subfield":{"id":"https://openalex.org/subfields/3104","display_name":"Condensed Matter Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10474","display_name":"Material Dynamics and Properties","score":0.009999999776482582,"subfield":{"id":"https://openalex.org/subfields/2505","display_name":"Materials Chemistry"},"field":{"id":"https://openalex.org/fields/25","display_name":"Materials Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11716","display_name":"Random Matrices and Applications","score":0.009600000455975533,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/sherrington-kirkpatrick-model","display_name":"Sherrington-Kirkpatrick model","score":0.9620000123977661},{"id":"https://openalex.org/keywords/spin-glass","display_name":"spin glass","score":0.734000027179718},{"id":"https://openalex.org/keywords/marginal-states","display_name":"marginal states","score":0.7390000224113464},{"id":"https://openalex.org/keywords/metastable-states","display_name":"metastable states","score":0.8740000128746033},{"id":"https://openalex.org/keywords/replica-symmetry-breaking","display_name":"replica symmetry breaking","score":0.8450000286102295},{"id":"https://openalex.org/keywords/annealed-approximation","display_name":"annealed approximation","score":0.7269999980926514},{"id":"https://openalex.org/keywords/equilibrium-states","display_name":"equilibrium states","score":0.5249999761581421}],"concepts":[{"id":"https://openalex.org/C150775274","wikidata":"https://www.wikidata.org/wiki/Q1490646","display_name":"Spin glass","level":2,"score":0.7990999817848206},{"id":"https://openalex.org/C89464430","wikidata":"https://www.wikidata.org/wiki/Q849516","display_name":"Metastability","level":2,"score":0.7684000134468079},{"id":"https://openalex.org/C2775937380","wikidata":"https://www.wikidata.org/wiki/Q1232589","display_name":"Replica","level":2,"score":0.7347999811172485},{"id":"https://openalex.org/C50488108","wikidata":"https://www.wikidata.org/wiki/Q618684","display_name":"Marginal stability","level":3,"score":0.6146000027656555},{"id":"https://openalex.org/C26445502","wikidata":"https://www.wikidata.org/wiki/Q503658","display_name":"Replica trick","level":3,"score":0.5842000246047974},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.5842000246047974},{"id":"https://openalex.org/C204795200","wikidata":"https://www.wikidata.org/wiki/Q903282","display_name":"Symmetry breaking","level":2,"score":0.49140000343322754},{"id":"https://openalex.org/C74859849","wikidata":"https://www.wikidata.org/wiki/Q2466686","display_name":"Non-equilibrium thermodynamics","level":2,"score":0.4068000018596649},{"id":"https://openalex.org/C26873012","wikidata":"https://www.wikidata.org/wiki/Q214781","display_name":"Condensed matter physics","level":1,"score":0.3828999996185303},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.37119999527931213},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.37119999527931213},{"id":"https://openalex.org/C186633575","wikidata":"https://www.wikidata.org/wiki/Q845060","display_name":"Maxima and minima","level":2,"score":0.3449000120162964},{"id":"https://openalex.org/C2780813799","wikidata":"https://www.wikidata.org/wiki/Q3274237","display_name":"Zero (linguistics)","level":2,"score":0.34150001406669617},{"id":"https://openalex.org/C2779886137","wikidata":"https://www.wikidata.org/wiki/Q21030012","display_name":"Symmetry (geometry)","level":2,"score":0.33970001339912415},{"id":"https://openalex.org/C75235859","wikidata":"https://www.wikidata.org/wiki/Q582659","display_name":"Exponential growth","level":2,"score":0.3255000114440918},{"id":"https://openalex.org/C186370098","wikidata":"https://www.wikidata.org/wiki/Q442787","display_name":"Energy (signal processing)","level":2,"score":0.3140999972820282},{"id":"https://openalex.org/C42704618","wikidata":"https://www.wikidata.org/wiki/Q910917","display_name":"Spin (aerodynamics)","level":2,"score":0.29409998655319214},{"id":"https://openalex.org/C84551667","wikidata":"https://www.wikidata.org/wiki/Q155640","display_name":"Potential energy","level":2,"score":0.2660999894142151},{"id":"https://openalex.org/C131157278","wikidata":"https://www.wikidata.org/wiki/Q51363","display_name":"Thermodynamic equilibrium","level":2,"score":0.25589999556541443},{"id":"https://openalex.org/C116674579","wikidata":"https://www.wikidata.org/wiki/Q193442","display_name":"Supersymmetry","level":2,"score":0.2508000135421753}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.04006","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.04006","pdf_url":"https://arxiv.org/pdf/2609.04006","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2609.04006","doi":"https://doi.org/10.48550/arxiv.2609.04006","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.04006","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.04006","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.04006","pdf_url":"https://arxiv.org/pdf/2609.04006","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320320300","display_name":"European Commission","ror":"https://ror.org/00k4n6c32"},{"id":"https://openalex.org/F4320338448","display_name":"HORIZON EUROPE European Innovation Council","ror":"https://ror.org/05cx8cy07"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7208838836.pdf","grobid_xml":"https://content.openalex.org/works/W7208838836.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W1784315482","https://openalex.org/W179161214","https://openalex.org/W1676358770","https://openalex.org/W6891961882","https://openalex.org/W2020997877","https://openalex.org/W2048717141","https://openalex.org/W1969416664","https://openalex.org/W2061706463","https://openalex.org/W2484013721","https://openalex.org/W3102509854"],"abstract_inverted_index":{"The":[0],"number":[1],"of":[2,5],"metastable":[3],"states":[4,23,43],"a":[6,53],"random":[7],"corrugated":[8],"landscape":[9],"controls":[10],"what":[11],"dynamics,":[12],"inference":[13],"and":[14,26,56,72],"learning":[15],"can":[16],"reach.":[17],"In":[18],"the":[19,31,37,48,65,75,80,87],"Sherrington-Kirkpatrick":[20],"model":[21],"these":[22],"are":[24],"marginal,":[25],"counting":[27],"them":[28],"requires":[29],"breaking":[30],"BRST":[32],"symmetry.":[33],"So":[34],"far":[35],"only":[36],"annealed":[38,66],"complexity":[39,50],"was":[40],"known,":[41],"predicting":[42],"below":[44],"equilibrium.":[45],"We":[46],"compute":[47],"quenched":[49],"function":[51],"with":[52],"two-group":[54],"Ansatz":[55],"one-step":[57],"replica":[58],"symmetry":[59],"breaking.":[60],"It":[61],"branches":[62],"continuously":[63],"off":[64],"curve":[67],"at":[68],"its":[69],"instability":[70],"point":[71],"vanishes":[73],"near":[74],"Parisi":[76],"free":[77],"energy,":[78],"while":[79],"soft":[81],"mode":[82],"turns":[83],"toward":[84],"orthogonality":[85],"to":[86],"magnetization.":[88]},"counts_by_year":[],"updated_date":"2026-10-28T10:33:24.094632","created_date":"2026-09-06T00:00:00"}