{"id":"https://openalex.org/W2291147962","doi":"https://doi.org/10.1137/15m1052081","title":"Pattern Size in Gaussian Fields from Spinodal Decomposition","display_name":"Pattern Size in Gaussian Fields from Spinodal Decomposition","publication_year":2017,"publication_date":"2017-01-01","ids":{"openalex":"https://openalex.org/W2291147962","doi":"https://doi.org/10.1137/15m1052081","mag":"2291147962"},"language":"en","primary_location":{"id":"doi:10.1137/15m1052081","is_oa":false,"landing_page_url":"https://doi.org/10.1137/15m1052081","pdf_url":null,"source":{"id":"https://openalex.org/S77200987","display_name":"SIAM Journal on Applied Mathematics","issn_l":"0036-1399","issn":["0036-1399","1095-712X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Applied Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1510.01967","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":null,"display_name":"Luigi Amedeo Bianchi","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Luigi Amedeo Bianchi","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":null,"display_name":"Dirk Bl\u00f6mker","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Dirk Bl\u00f6mker","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":null,"display_name":"Philipp Wacker","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Philipp Wacker","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.2057,"has_fulltext":true,"cited_by_count":3,"citation_normalized_percentile":{"value":0.4229262,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":94,"max":97},"biblio":{"volume":"77","issue":"4","first_page":"1292","last_page":"1319"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11087","display_name":"Solidification and crystal growth phenomena","score":0.7644000053405762,"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"}},"topics":[{"id":"https://openalex.org/T11087","display_name":"Solidification and crystal growth phenomena","score":0.7644000053405762,"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/T11152","display_name":"Stochastic processes and statistical mechanics","score":0.04270000010728836,"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"}},{"id":"https://openalex.org/T12141","display_name":"Fluid Dynamics and Thin Films","score":0.02329999953508377,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/eigenfunction","display_name":"Eigenfunction","score":0.6460000276565552},{"id":"https://openalex.org/keywords/spinodal-decomposition","display_name":"Spinodal decomposition","score":0.6212999820709229},{"id":"https://openalex.org/keywords/ergodic-theory","display_name":"Ergodic theory","score":0.573199987411499},{"id":"https://openalex.org/keywords/gaussian","display_name":"Gaussian","score":0.5393999814987183},{"id":"https://openalex.org/keywords/fourier-series","display_name":"Fourier series","score":0.4823000133037567},{"id":"https://openalex.org/keywords/fourier-transform","display_name":"Fourier transform","score":0.47130000591278076},{"id":"https://openalex.org/keywords/gaussian-random-field","display_name":"Gaussian random field","score":0.4575999975204468},{"id":"https://openalex.org/keywords/random-field","display_name":"Random field","score":0.43209999799728394},{"id":"https://openalex.org/keywords/laplace-operator","display_name":"Laplace operator","score":0.4251999855041504},{"id":"https://openalex.org/keywords/series","display_name":"Series (stratigraphy)","score":0.41909998655319214}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7437000274658203},{"id":"https://openalex.org/C128803854","wikidata":"https://www.wikidata.org/wiki/Q1307821","display_name":"Eigenfunction","level":3,"score":0.6460000276565552},{"id":"https://openalex.org/C29234059","wikidata":"https://www.wikidata.org/wiki/Q2666979","display_name":"Spinodal decomposition","level":3,"score":0.6212999820709229},{"id":"https://openalex.org/C122044880","wikidata":"https://www.wikidata.org/wiki/Q5498822","display_name":"Ergodic theory","level":2,"score":0.573199987411499},{"id":"https://openalex.org/C163716315","wikidata":"https://www.wikidata.org/wiki/Q901177","display_name":"Gaussian","level":2,"score":0.5393999814987183},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5045999884605408},{"id":"https://openalex.org/C207864730","wikidata":"https://www.wikidata.org/wiki/Q179467","display_name":"Fourier series","level":2,"score":0.4823000133037567},{"id":"https://openalex.org/C102519508","wikidata":"https://www.wikidata.org/wiki/Q6520159","display_name":"Fourier transform","level":2,"score":0.47130000591278076},{"id":"https://openalex.org/C51267290","wikidata":"https://www.wikidata.org/wiki/Q5527848","display_name":"Gaussian random field","level":4,"score":0.4575999975204468},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.45080000162124634},{"id":"https://openalex.org/C130402806","wikidata":"https://www.wikidata.org/wiki/Q5361768","display_name":"Random field","level":2,"score":0.43209999799728394},{"id":"https://openalex.org/C165700671","wikidata":"https://www.wikidata.org/wiki/Q203484","display_name":"Laplace operator","level":2,"score":0.4251999855041504},{"id":"https://openalex.org/C143724316","wikidata":"https://www.wikidata.org/wiki/Q312468","display_name":"Series (stratigraphy)","level":2,"score":0.41909998655319214},{"id":"https://openalex.org/C178009071","wikidata":"https://www.wikidata.org/wiki/Q93344","display_name":"Trigonometric functions","level":2,"score":0.3537999987602234},{"id":"https://openalex.org/C198082294","wikidata":"https://www.wikidata.org/wiki/Q3399648","display_name":"Position (finance)","level":2,"score":0.3449999988079071},{"id":"https://openalex.org/C203024314","wikidata":"https://www.wikidata.org/wiki/Q1365258","display_name":"Fourier analysis","level":3,"score":0.3425999879837036},{"id":"https://openalex.org/C9652623","wikidata":"https://www.wikidata.org/wiki/Q190109","display_name":"Field (mathematics)","level":2,"score":0.33480000495910645},{"id":"https://openalex.org/C192209626","wikidata":"https://www.wikidata.org/wiki/Q190909","display_name":"Focus (optics)","level":2,"score":0.3264999985694885},{"id":"https://openalex.org/C99730327","wikidata":"https://www.wikidata.org/wiki/Q1501536","display_name":"Sobolev space","level":2,"score":0.32249999046325684},{"id":"https://openalex.org/C151342819","wikidata":"https://www.wikidata.org/wiki/Q62542","display_name":"Phase space","level":2,"score":0.3151000142097473},{"id":"https://openalex.org/C121194460","wikidata":"https://www.wikidata.org/wiki/Q856741","display_name":"Random walk","level":2,"score":0.3138999938964844},{"id":"https://openalex.org/C61326573","wikidata":"https://www.wikidata.org/wiki/Q1496376","display_name":"Gaussian process","level":3,"score":0.310699999332428},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.2989000082015991},{"id":"https://openalex.org/C201779956","wikidata":"https://www.wikidata.org/wiki/Q5426803","display_name":"Ergodicity","level":2,"score":0.2976999878883362},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.289000004529953},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.28630000352859497},{"id":"https://openalex.org/C112401455","wikidata":"https://www.wikidata.org/wiki/Q178036","display_name":"Brownian motion","level":2,"score":0.28380000591278076},{"id":"https://openalex.org/C91188154","wikidata":"https://www.wikidata.org/wiki/Q186247","display_name":"Vector field","level":2,"score":0.2824999988079071},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.2784000039100647},{"id":"https://openalex.org/C186080144","wikidata":"https://www.wikidata.org/wiki/Q358733","display_name":"Series expansion","level":2,"score":0.27630001306533813},{"id":"https://openalex.org/C8272713","wikidata":"https://www.wikidata.org/wiki/Q176737","display_name":"Stochastic process","level":2,"score":0.2705000042915344},{"id":"https://openalex.org/C2778755073","wikidata":"https://www.wikidata.org/wiki/Q10858537","display_name":"Scale (ratio)","level":2,"score":0.267300009727478},{"id":"https://openalex.org/C27156116","wikidata":"https://www.wikidata.org/wiki/Q1778098","display_name":"Pointwise convergence","level":3,"score":0.2651999890804291},{"id":"https://openalex.org/C5806529","wikidata":"https://www.wikidata.org/wiki/Q2365325","display_name":"Orthonormal basis","level":2,"score":0.25609999895095825}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1137/15m1052081","is_oa":false,"landing_page_url":"https://doi.org/10.1137/15m1052081","pdf_url":null,"source":{"id":"https://openalex.org/S77200987","display_name":"SIAM Journal on Applied Mathematics","issn_l":"0036-1399","issn":["0036-1399","1095-712X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Applied Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1510.01967","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1510.01967","pdf_url":"https://arxiv.org/pdf/1510.01967","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,"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":"pmh:oai:iris.unitn.it:11572/244020","is_oa":true,"landing_page_url":"https://epubs.siam.org/doi/epdf/10.1137/24M1690631","pdf_url":"https://iris.unitn.it/bitstream/11572/244020/2/BiaBloWacPattern.pdf","source":{"id":"https://openalex.org/S4377196320","display_name":"Iris (University of Trento)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I193223587","host_organization_name":"University of Trento","host_organization_lineage":["https://openalex.org/I193223587"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"info:eu-repo/semantics/article"},{"id":"pmh:oai:uni-augsburg.opus-bayern.de:43678","is_oa":false,"landing_page_url":"https://nbn-resolving.org/urn:nbn:de:bvb:384-opus4-436781","pdf_url":null,"source":{"id":"https://openalex.org/S4306400930","display_name":"OPUS (Augsburg University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I119916105","host_organization_name":"Augsburg University","host_organization_lineage":["https://openalex.org/I119916105"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"article"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1510.01967","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1510.01967","pdf_url":"https://arxiv.org/pdf/1510.01967","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,"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":[],"awards":[{"id":"https://openalex.org/G8808630263","display_name":null,"funder_award_id":"BL 535/9-2","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"}],"funders":[{"id":"https://openalex.org/F4320320879","display_name":"Deutsche Forschungsgemeinschaft","ror":"https://ror.org/018mejw64"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":29,"referenced_works":["https://openalex.org/W1738066432","https://openalex.org/W1966291642","https://openalex.org/W1972931004","https://openalex.org/W1977895317","https://openalex.org/W1990010228","https://openalex.org/W2001503165","https://openalex.org/W2005581903","https://openalex.org/W2006214482","https://openalex.org/W2019950972","https://openalex.org/W2031698575","https://openalex.org/W2034549893","https://openalex.org/W2041646860","https://openalex.org/W2046528357","https://openalex.org/W2049703307","https://openalex.org/W2053812019","https://openalex.org/W2054549611","https://openalex.org/W2061344677","https://openalex.org/W2061444804","https://openalex.org/W2065874007","https://openalex.org/W2066985259","https://openalex.org/W2072863189","https://openalex.org/W2081963949","https://openalex.org/W2086126894","https://openalex.org/W2115189693","https://openalex.org/W2165190032","https://openalex.org/W2252910020","https://openalex.org/W2320840070","https://openalex.org/W2357114303","https://openalex.org/W2593708767"],"related_works":[],"abstract_inverted_index":{"We":[0],"study":[1,75,169],"the":[2,9,21,34,42,76,85,105,108,118,151,170,178,187,203,206,212],"two-dimensional":[3],"snake-like":[4],"pattern":[5,171],"that":[6,127,201,211],"arises":[7],"at":[8,160],"onset":[10],"of":[11,14,33,41,79,136,173,180,205,215,219],"phase":[12],"separation":[13],"alloys":[15],"described":[16],"by":[17,95,193],"spinodal":[18,80],"decomposition":[19,81],"in":[20,52,64,70,107,124],"Cahn--Hilliard":[22,109],"model.":[23,110],"These":[24],"are":[25,39],"somewhat":[26],"universal":[27],"patterns":[28,69],"due":[29],"to":[30,74,146],"an":[31],"overlay":[32],"most":[35],"unstable":[36],"pattern,":[37],"which":[38,103],"eigenfunctions":[40],"Laplacian":[43],"all":[44],"with":[45,99,133,222],"a":[46,114,134,140,157,161,191,223],"similar":[47],"wave-number.":[48],"Similar":[49],"structures":[50],"appear":[51],"other":[53],"models":[54],"like":[55],"reaction-diffusion":[56],"systems":[57],"describing":[58],"animal":[59],"coats'":[60],"patterns,":[61],"hill":[62],"formation":[63],"surface":[65],"growth,":[66],"or":[67],"vegetation":[68],"desertification.":[71],"In":[72],"order":[73,137,220],"early":[77],"stages":[78],"we":[82,176,209],"focus":[83],"on":[84,182],"linearized":[86],"equation.":[87],"Our":[88],"main":[89],"result":[90],"studies":[91],"random":[92,158],"functions":[93],"given":[94,225],"cosine":[96,115],"Fourier":[97,125],"series":[98],"independent":[100],"Gaussian":[101],"coefficients,":[102],"dominate":[104],"dynamics":[106],"This":[111],"is":[112,120,155,217],"not":[113,128],"process,":[116],"as":[117,156],"sum":[119],"taken":[121],"over":[122],"domains":[123,175],"space":[126],"only":[129],"grow":[130],"and":[131,195,197],"scale":[132],"parameter":[135],"$1/\\varepsilon$":[138],"for":[139,148],"small":[141],"$0<\\varepsilon\\ll1$,":[142],"but":[143],"also":[144],"move":[145],"infinity":[147],"$\\varepsilon\\to0$.":[149],"Moreover,":[150],"model":[152],"under":[153],"consideration":[154],"field":[159],"fixed":[162],"time":[163],"neither":[164],"stationary":[165],"nor":[166],"isotropic.":[167],"To":[168],"size":[172],"nodal":[174],"consider":[177],"density":[179],"zeros":[181,216],"any":[183],"straight":[184],"line":[185],"through":[186],"spatial":[188],"domain.":[189],"Using":[190],"theorem":[192],"Edelman":[194],"Kostlan":[196],"weighted":[198],"ergodic":[199],"theorems":[200],"ensure":[202],"convergence":[204],"moving":[207],"sums,":[208],"show":[210],"average":[213],"distance":[214],"asymptotically":[218],"$\\varepsilon$":[221],"precisely":[224],"constant.":[226]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2020,"cited_by_count":2}],"updated_date":"2026-07-28T07:46:37.118299","created_date":"2016-06-24T00:00:00"}