{"id":"https://openalex.org/W7215940936","doi":"https://doi.org/10.48550/arxiv.2610.00011","title":"Critical Free-Energy Variance in the Sherrington-Kirkpatrick Model","display_name":"Critical Free-Energy Variance in the Sherrington-Kirkpatrick Model","publication_year":2026,"publication_date":"2026-07-09","ids":{"openalex":"https://openalex.org/W7215940936","doi":"https://doi.org/10.48550/arxiv.2610.00011"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.00011","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.00011","pdf_url":"https://arxiv.org/pdf/2610.00011","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2610.00011","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5082850015","display_name":"Miguel Tierz","orcid":"https://orcid.org/0000-0002-4579-4364","observed_orcids":["https://orcid.org/0000-0002-4579-4364"]},"institutions":[{"id":"https://openalex.org/I4403928334","display_name":"Shanghai Institute for Mathematics and Interdisciplinary Sciences","ror":"https://ror.org/0489cd473","country_code":"CN","type":"government","lineage":["https://openalex.org/I4403928334"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Tierz, Miguel","raw_affiliation_strings":["Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS) , Block A , No. 657 Songhu Road , Yangpu District , Shanghai , China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS) , Block A , No. 657 Songhu Road , Yangpu District , Shanghai , China","institution_ids":["https://openalex.org/I4403928334"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5082850015"],"corresponding_institution_ids":["https://openalex.org/I4403928334"],"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/T11716","display_name":"Random Matrices and Applications","score":0.8036999702453613,"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"}},"topics":[{"id":"https://openalex.org/T11716","display_name":"Random Matrices and Applications","score":0.8036999702453613,"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"}},{"id":"https://openalex.org/T10591","display_name":"Theoretical and Computational Physics","score":0.14169999957084656,"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/T11152","display_name":"Stochastic processes and statistical mechanics","score":0.014700000174343586,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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":1.0},{"id":"https://openalex.org/keywords/spin-glass","display_name":"spin glass","score":0.7459999918937683},{"id":"https://openalex.org/keywords/spin-glass-susceptibility","display_name":"spin-glass susceptibility","score":0.8130000233650208},{"id":"https://openalex.org/keywords/critical-points","display_name":"critical points","score":0.796999990940094},{"id":"https://openalex.org/keywords/finite-size-effects","display_name":"finite-size effects","score":0.4169999957084656}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8467000126838684},{"id":"https://openalex.org/C39613435","wikidata":"https://www.wikidata.org/wiki/Q846677","display_name":"Remainder","level":2,"score":0.8342999815940857},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.7402999997138977},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.6991999745368958},{"id":"https://openalex.org/C196083921","wikidata":"https://www.wikidata.org/wiki/Q7915758","display_name":"Variance (accounting)","level":2,"score":0.6589999794960022},{"id":"https://openalex.org/C99844830","wikidata":"https://www.wikidata.org/wiki/Q102441924","display_name":"Scaling","level":2,"score":0.5777999758720398},{"id":"https://openalex.org/C2778355321","wikidata":"https://www.wikidata.org/wiki/Q17079427","display_name":"Identity (music)","level":2,"score":0.5166000127792358},{"id":"https://openalex.org/C196298200","wikidata":"https://www.wikidata.org/wiki/Q577705","display_name":"Critical point (mathematics)","level":2,"score":0.4675000011920929},{"id":"https://openalex.org/C28719098","wikidata":"https://www.wikidata.org/wiki/Q44946","display_name":"Point (geometry)","level":2,"score":0.4278999865055084},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4237000048160553},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.36649999022483826},{"id":"https://openalex.org/C186370098","wikidata":"https://www.wikidata.org/wiki/Q442787","display_name":"Energy (signal processing)","level":2,"score":0.3637000024318695},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.3582000136375427},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.34470000863075256},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.323199987411499},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.31360000371932983},{"id":"https://openalex.org/C2778755073","wikidata":"https://www.wikidata.org/wiki/Q10858537","display_name":"Scale (ratio)","level":2,"score":0.2973000109195709},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.27970001101493835},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.25929999351501465},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.2547999918460846},{"id":"https://openalex.org/C61445026","wikidata":"https://www.wikidata.org/wiki/Q217608","display_name":"Fixed point","level":2,"score":0.25130000710487366}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.00011","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.00011","pdf_url":"https://arxiv.org/pdf/2610.00011","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2610.00011","doi":"https://doi.org/10.48550/arxiv.2610.00011","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.00011","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":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2610.00011","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.00011","pdf_url":"https://arxiv.org/pdf/2610.00011","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7215940936.pdf","grobid_xml":"https://content.openalex.org/works/W7215940936.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W7165656901","https://openalex.org/W7167240993","https://openalex.org/W2971722084","https://openalex.org/W7170449402","https://openalex.org/W4406733252","https://openalex.org/W7212072076","https://openalex.org/W7223557182","https://openalex.org/W1637762409","https://openalex.org/W7095505633","https://openalex.org/W7134259422"],"abstract_inverted_index":{"We":[0,126],"study":[1],"the":[2,5,9,36,52,66,107,115,120],"variance":[3,121],"of":[4],"free":[6],"energy":[7],"in":[8],"Gaussian":[10,61],"Sherrington-Kirkpatrick":[11],"model":[12],"at":[13],"zero":[14],"external":[15],"field.":[16],"For":[17],"every":[18],"system":[19],"size":[20],"and":[21,39,60,93,112,136],"inverse":[22],"temperature":[23,134],"$\u03b2>0$,":[24],"we":[25],"prove":[26],"$2N^2\\operatorname{Var}":[27,70],"F_N^{\\mathrm{off}}(\u03b2)":[28],"\\le":[29],"\\log\u03c7+(\u03b2^2-1)(\u03c7-1),":[30],"\u03c7=N\\mathbb{E}\\langle":[31],"R_{1,2}^2\\rangle_\u03b2$,":[32],"where":[33],"$R_{1,2}$":[34],"is":[35,101],"replica":[37],"overlap":[38,89,138],"$F_N^{\\mathrm{off}}=N^{-1}\\log":[40],"Z_N^{\\mathrm{off}}$":[41],"omits":[42],"diagonal":[43],"disorder.":[44],"The":[45],"proof":[46],"uses":[47],"an":[48,128],"exact":[49],"identity":[50],"for":[51],"edges":[53],"incident":[54],"to":[55,69,85],"one":[56],"spin,":[57],"Schur":[58],"complements,":[59],"disorder":[62],"interpolation.":[63],"At":[64],"criticality":[65],"bound":[67,130],"reduces":[68],"F_N^{\\mathrm{off}}(1)\\le\\log\u03c7$.":[71],"An":[72],"explicit":[73],"nonnegative":[74],"integral":[75,100],"measures":[76],"its":[77,98],"gap.":[78],"A":[79],"second":[80],"decomposition":[81],"relates":[82],"this":[83],"gap":[84,116],"a":[86,123,142],"mixed":[87],"cubic":[88],"functional.":[90],"Existing":[91],"moment":[92,139],"cavity":[94],"estimates":[95],"imply":[96],"that":[97,114],"required":[99],"bounded.":[102],"Critical":[103],"susceptibility":[104],"asymptotics":[105],"give":[106],"sharp":[108],"logarithmic":[109],"upper":[110],"coefficient":[111],"show":[113],"converges":[117],"exactly":[118],"when":[119],"has":[122],"constant":[124],"term.":[125],"extend":[127],"entropy":[129],"throughout":[131],"bounded":[132],"critical":[133],"windows":[135],"establish":[137],"stability":[140],"on":[141],"smaller":[143],"interval.":[144]},"counts_by_year":[],"updated_date":"2026-11-06T12:05:46.655746","created_date":"2026-10-03T00:00:00"}