{"id":"https://openalex.org/W7225622799","doi":"https://doi.org/10.5281/zenodo.24069926","title":"The second largest outdegree of conditioned Galton\u2013Watson trees with infinite variance","display_name":"The second largest outdegree of conditioned Galton\u2013Watson trees with infinite variance","publication_year":2026,"publication_date":"2026-11-08","ids":{"openalex":"https://openalex.org/W7225622799","doi":"https://doi.org/10.5281/zenodo.24069926"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.24069926","doi":"https://doi.org/10.5281/zenodo.24069926","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24069926","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.5281/zenodo.24069926","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5041315596","display_name":"Yitzchak Shmalo","orcid":"https://orcid.org/0000-0002-2886-3049","observed_orcids":["https://orcid.org/0000-0002-2886-3049"]},"institutions":[{"id":"https://openalex.org/I197251160","display_name":"Hebrew University of Jerusalem","ror":"https://ror.org/03qxff017","country_code":"IL","type":"education","lineage":["https://openalex.org/I197251160"]}],"countries":["IL"],"is_corresponding":true,"raw_author_name":"Shmalo, Yitzchak","raw_affiliation_strings":["Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel","institution_ids":["https://openalex.org/I197251160"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5041315596"],"corresponding_institution_ids":["https://openalex.org/I197251160"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"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/T11152","display_name":"Stochastic processes and statistical mechanics","score":0.9961000084877014,"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"}},"topics":[{"id":"https://openalex.org/T11152","display_name":"Stochastic processes and statistical mechanics","score":0.9961000084877014,"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.0006000000284984708,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"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/T13500","display_name":"Probability and Statistical Research","score":0.00039999998989515007,"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/galton-watson-trees","display_name":"Galton-Watson trees","score":0.9089999794960022},{"id":"https://openalex.org/keywords/out-degree","display_name":"out-degree","score":0.6069999933242798},{"id":"https://openalex.org/keywords/order-statistics","display_name":"order statistics","score":0.5450000166893005},{"id":"https://openalex.org/keywords/counterexample","display_name":"counterexample","score":0.6050000190734863}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8417999744415283},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7300999760627747},{"id":"https://openalex.org/C113174947","wikidata":"https://www.wikidata.org/wiki/Q2859736","display_name":"Tree (set theory)","level":2,"score":0.5282999873161316},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.5062999725341797},{"id":"https://openalex.org/C18653775","wikidata":"https://www.wikidata.org/wiki/Q1333358","display_name":"Joint probability distribution","level":2,"score":0.49619999527931213},{"id":"https://openalex.org/C122123141","wikidata":"https://www.wikidata.org/wiki/Q176623","display_name":"Random variable","level":2,"score":0.4844000041484833},{"id":"https://openalex.org/C110121322","wikidata":"https://www.wikidata.org/wiki/Q865811","display_name":"Distribution (mathematics)","level":2,"score":0.43209999799728394},{"id":"https://openalex.org/C149441793","wikidata":"https://www.wikidata.org/wiki/Q200726","display_name":"Probability distribution","level":2,"score":0.427700012922287},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.421099990606308},{"id":"https://openalex.org/C196083921","wikidata":"https://www.wikidata.org/wiki/Q7915758","display_name":"Variance (accounting)","level":2,"score":0.3930000066757202},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.35350000858306885},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3212999999523163},{"id":"https://openalex.org/C182365436","wikidata":"https://www.wikidata.org/wiki/Q50701","display_name":"Variable (mathematics)","level":2,"score":0.25999999046325684}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.5281/zenodo.24069926","doi":"https://doi.org/10.5281/zenodo.24069926","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24069926","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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"},{"id":"doi:10.5281/zenodo.24069927","doi":"https://doi.org/10.5281/zenodo.24069927","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24069927","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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":"doi:10.5281/zenodo.24069926","doi":"https://doi.org/10.5281/zenodo.24069926","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24069926","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W2257606390","https://openalex.org/W2592628246","https://openalex.org/W7169547041","https://openalex.org/W2904103226","https://openalex.org/W104561473","https://openalex.org/W1974390789","https://openalex.org/W2158106435","https://openalex.org/W2006252824","https://openalex.org/W2027748879","https://openalex.org/W2148672712"],"abstract_inverted_index":{"Let":[0],"$\\xi$":[1,29,52],"be":[2,16,38],"a":[3,23],"random":[4,217],"variable":[5],"with":[6,26,85,96,130,141,148,156],"values":[7],"in":[8,19],"$\\{0,1,2,\\dots\\}$":[9],"and":[10,35,62,92,108,133,136,139,192,202],"$\\mathbb":[11],"E\\xi=1$,":[12],"let":[13,36],"$Y_{(1)}\\ge":[14],"Y_{(2)}\\ge\\dots$":[15],"the":[17,39,66,79,112,119,153,183,193,205,216],"outdegrees,":[18],"decreasing":[20],"order,":[21],"of":[22,42,46,74,163,168,187,197,204,208,219],"Galton\u2013Watson":[24],"tree":[25,155],"offspring":[27,128],"distribution":[28,129],"conditioned":[30,154],"to":[31],"have":[32,64,175],"$n$":[33,43,223],"vertices,":[34],"$\\xi_{(1)}\\ge\\xi_{(2)}\\ge\\dots$":[37],"order":[40,189,199],"statistics":[41],"independent":[44],"copies":[45],"$\\xi$.":[47],"Janson":[48],"proved":[49],"that":[50,83,93,111,118,147],"if":[51],"has":[53,159],"finite":[54],"variance":[55,80],"then,":[56],"for":[57,99,215],"every":[58,100],"fixed":[59,101],"$j$,":[60],"$Y_{(j)}$":[61],"$\\xi_{(j)}$":[63],"asymptotically":[65],"same":[67,213],"distribution.":[68],"He":[69],"asked":[70],"whether":[71],"two":[72],"consequences":[73],"this":[75],"remain":[76],"true":[77],"when":[78,88,103],"is":[81,186,196],"infinite:":[82],"$Y_{(1)}<h(n)$":[84],"high":[86,97],"probability":[87,98,149],"$n\\mathbb":[89,104],"P(\\xi\\ge":[90,105,143],"h(n))\\to0$,":[91],"$Y_{(j)}\\ge":[94],"h(n)$":[95],"$j$":[102],"h(n))\\to\\infty$.":[106],"Kortchemski":[107],"Richier":[109],"showed":[110],"first":[113],"can":[114,121],"fail.":[115],"We":[116,125],"show":[117],"second":[120,194],"fail":[122],"as":[123],"well.":[124],"construct":[126],"an":[127],"mean":[131],"$1$":[132],"infinite":[134],"variance,":[135],"integers":[137],"$n_r\\to\\infty$":[138],"$L_r$":[140],"$n_r\\mathbb":[142],"L_r)\\ge":[144],"r$,":[145],"such":[146],"at":[150,165,177],"least":[151,166],"$1-1/r$":[152],"$n_r$":[157,182],"vertices":[158,174],"exactly":[160],"one":[161],"vertex":[162],"outdegree":[164,169,176,185],"$L_r$,":[167],"$(r+\\frac12)L_r$,":[170],"while":[171],"all":[172],"other":[173],"most":[178],"$L_r/(64(\\log":[179],"n_r)^2)$.":[180],"Along":[181],"largest":[184,195],"larger":[188],"than":[190,200],"$\\xi_{(1)}$":[191],"smaller":[198],"$\\xi_{(2)}$,":[201],"each":[203],"six":[206],"conclusions":[207],"Janson's":[209],"theorem":[210],"fails.":[211],"The":[212],"holds":[214],"allocation":[218],"$n-1$":[220],"balls":[221],"into":[222],"boxes.":[224]},"counts_by_year":[],"updated_date":"2026-11-11T12:16:26.678958","created_date":"2026-11-10T00:00:00"}