{"id":"https://openalex.org/W2884677828","doi":"https://doi.org/10.48550/arxiv.1807.03775","title":"Probability laws for the distribution of geometric lengths when sampling by a random walk in a Fuchsian fundamental group","display_name":"Probability laws for the distribution of geometric lengths when sampling by a random walk in a Fuchsian fundamental group","publication_year":2018,"publication_date":"2018-07-10","ids":{"openalex":"https://openalex.org/W2884677828","doi":"https://doi.org/10.48550/arxiv.1807.03775","mag":"2884677828"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:1807.03775","doi":null,"is_oa":true,"landing_page_url":"http://arxiv.org/abs/1807.03775","pdf_url":"https://arxiv.org/pdf/1807.03775","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/1807.03775","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5087342854","display_name":"Peter Seho Park","orcid":"https://orcid.org/0000-0002-6532-0529","observed_orcids":["https://orcid.org/0000-0002-6532-0529"]},"institutions":[{"id":"https://openalex.org/I136199984","display_name":"Harvard University","ror":"https://ror.org/03vek6s52","country_code":"US","type":"education","lineage":["https://openalex.org/I136199984"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Park, Peter S.","raw_affiliation_strings":["Department of Mathematics , Harvard University , Cambridge , MA 02138"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , Harvard University , Cambridge , MA 02138","institution_ids":["https://openalex.org/I136199984"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5087342854"],"corresponding_institution_ids":["https://openalex.org/I136199984"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":2,"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/T10304","display_name":"Geometric and Algebraic Topology","score":0.6553000211715698,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/T10304","display_name":"Geometric and Algebraic Topology","score":0.6553000211715698,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.2639000117778778,"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/T10229","display_name":"Geometric Analysis and Curvature Flows","score":0.015399999916553497,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/random-walks-on-groups","display_name":"random walks on groups","score":0.8880000114440918},{"id":"https://openalex.org/keywords/fuchsian-groups","display_name":"Fuchsian groups","score":0.8960000276565552},{"id":"https://openalex.org/keywords/hyperbolic-surfaces","display_name":"hyperbolic surfaces","score":0.9660000205039978},{"id":"https://openalex.org/keywords/fundamental-group","display_name":"fundamental group","score":0.6439999938011169},{"id":"https://openalex.org/keywords/hyperbolic-elements","display_name":"hyperbolic elements","score":0.7720000147819519},{"id":"https://openalex.org/keywords/translation-length","display_name":"translation length","score":0.6489999890327454},{"id":"https://openalex.org/keywords/gromov-s-theorem","display_name":"Gromov's theorem","score":0.6970000267028809},{"id":"https://openalex.org/keywords/law-of-large-numbers","display_name":"law of large numbers","score":0.9129999876022339},{"id":"https://openalex.org/keywords/central-limit-theorem","display_name":"central limit theorem","score":0.9990000128746033},{"id":"https://openalex.org/keywords/large-deviations-principle","display_name":"large deviations principle","score":0.8840000033378601},{"id":"https://openalex.org/keywords/local-limit-theorem","display_name":"local limit theorem","score":0.9950000047683716},{"id":"https://openalex.org/keywords/probability-laws","display_name":"probability laws","score":0.3319999873638153}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8087674379348755},{"id":"https://openalex.org/C2780345915","wikidata":"https://www.wikidata.org/wiki/Q11185","display_name":"Backslash","level":2,"score":0.7214425802230835},{"id":"https://openalex.org/C13862481","wikidata":"https://www.wikidata.org/wiki/Q205124","display_name":"Fuchsian group","level":2,"score":0.6341997981071472},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.567672610282898},{"id":"https://openalex.org/C110121322","wikidata":"https://www.wikidata.org/wiki/Q865811","display_name":"Distribution (mathematics)","level":2,"score":0.563277542591095},{"id":"https://openalex.org/C2781311116","wikidata":"https://www.wikidata.org/wiki/Q83306","display_name":"Group (periodic table)","level":2,"score":0.49407675862312317},{"id":"https://openalex.org/C21031990","wikidata":"https://www.wikidata.org/wiki/Q355020","display_name":"Probability measure","level":2,"score":0.4854908287525177},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.48101621866226196},{"id":"https://openalex.org/C2780009758","wikidata":"https://www.wikidata.org/wiki/Q6804172","display_name":"Measure (data warehouse)","level":2,"score":0.47954225540161133},{"id":"https://openalex.org/C166785042","wikidata":"https://www.wikidata.org/wiki/Q190391","display_name":"Central limit theorem","level":2,"score":0.4753246605396271},{"id":"https://openalex.org/C121194460","wikidata":"https://www.wikidata.org/wiki/Q856741","display_name":"Random walk","level":2,"score":0.4580177366733551},{"id":"https://openalex.org/C149441793","wikidata":"https://www.wikidata.org/wiki/Q200726","display_name":"Probability distribution","level":2,"score":0.4206700325012207},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.35276076197624207},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.34384632110595703},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3356380760669708},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.10177281498908997},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.07967033982276917},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0},{"id":"https://openalex.org/C77088390","wikidata":"https://www.wikidata.org/wiki/Q8513","display_name":"Database","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"pmh:oai:arXiv.org:1807.03775","doi":null,"is_oa":true,"landing_page_url":"http://arxiv.org/abs/1807.03775","pdf_url":"https://arxiv.org/pdf/1807.03775","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":"mag:2884677828","doi":null,"is_oa":true,"landing_page_url":"http://arxiv.org/pdf/1807.03775.pdf","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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"arXiv (Cornell University)","raw_type":null},{"id":"doi:10.48550/arxiv.1807.03775","doi":"https://doi.org/10.48550/arxiv.1807.03775","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1807.03775","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:1807.03775","doi":null,"is_oa":true,"landing_page_url":"http://arxiv.org/abs/1807.03775","pdf_url":"https://arxiv.org/pdf/1807.03775","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":[{"id":"https://openalex.org/G4973792023","display_name":"Graduate Research Fellowship Program (GRFP)","funder_award_id":"1745303","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2884677828.pdf","grobid_xml":"https://content.openalex.org/works/W2884677828.grobid-xml"},"referenced_works_count":14,"referenced_works":["https://openalex.org/W43328240","https://openalex.org/W1495928998","https://openalex.org/W1996046995","https://openalex.org/W2042920539","https://openalex.org/W2085797473","https://openalex.org/W2086605818","https://openalex.org/W2122542302","https://openalex.org/W2157174257","https://openalex.org/W2330852812","https://openalex.org/W2481146968","https://openalex.org/W2570329849","https://openalex.org/W2788187281","https://openalex.org/W2888037764","https://openalex.org/W2963578987"],"related_works":["https://openalex.org/W2844508005","https://openalex.org/W3175201585","https://openalex.org/W3182630221","https://openalex.org/W2037333982","https://openalex.org/W3047134159","https://openalex.org/W2307874459","https://openalex.org/W3136753903","https://openalex.org/W1885344917","https://openalex.org/W3189544457","https://openalex.org/W4411339431"],"abstract_inverted_index":{"Let":[0],"$S=\u0393\\backslash":[1],"\\mathbb{H}$":[2],"be":[3],"a":[4,76,93],"hyperbolic":[5,58],"surface":[6],"of":[7,27,52,57,72,78,95,105],"finite":[8],"topological":[9],"type,":[10],"such":[11],"that":[12],"the":[13,50,70],"Fuchsian":[14],"group":[15],"$\u0393\\le":[16],"\\operatorname{PSL}_2(\\mathbb{R})$":[17],"is":[18],"non-elementary,":[19],"and":[20,66,87],"consider":[21],"any":[22],"generating":[23],"set":[24,55],"$\\mathfrak":[25,48],"S$":[26],"$\u0393$.":[28],"When":[29],"sampling":[30],"by":[31,44],"an":[32,45],"$n$-step":[33],"random":[34],"walk":[35],"in":[36,47],"$\u03c0_1(S)":[37],"\\cong":[38],"\u0393$":[39],"with":[40],"each":[41],"step":[42],"given":[43],"element":[46],"S$,":[49],"subset":[51],"this":[53,68,96],"sampled":[54],"comprised":[56],"elements":[59],"approaches":[60],"full":[61],"measure":[62],"as":[63],"$n\\to":[64],"\\infty$,":[65],"for":[67],"subset,":[69],"distribution":[71],"geometric":[73],"lengths":[74,104],"obeys":[75],"Law":[77],"Large":[79,84],"Numbers,":[80],"Central":[81],"Limit":[82,89],"Theorem,":[83],"Deviations":[85],"Principle,":[86],"Local":[88],"Theorem.":[90],"We":[91],"give":[92],"proof":[94],"known":[97],"theorem":[98,101],"using":[99],"Gromov's":[100],"on":[102],"translation":[103],"Gromov-hyperbolic":[106],"groups.":[107]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2019,"cited_by_count":1}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2018-08-03T00:00:00"}