{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,29]],"date-time":"2026-01-29T13:33:24Z","timestamp":1769693604402,"version":"3.49.0"},"reference-count":27,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2023,11,20]],"date-time":"2023-11-20T00:00:00Z","timestamp":1700438400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2024,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>A spread-out lattice animal is a finite connected set of edges in <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832300038X_inline2.png\"\/><jats:tex-math>$\\{\\{x,y\\}\\subset \\mathbb{Z}^d\\;:\\;0\\lt \\|x-y\\|\\le L\\}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. A lattice tree is a lattice animal with no loops. The best estimate on the critical point <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832300038X_inline3.png\"\/><jats:tex-math>$p_{\\textrm{c}}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> so far was achieved by Penrose (<jats:italic>J. Stat. Phys.<\/jats:italic> 77, 3\u201315, 1994) : <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832300038X_inline4.png\"\/><jats:tex-math>$p_{\\textrm{c}}=1\/e+O(L^{-2d\/7}\\log L)$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> for both models for all <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832300038X_inline5.png\"\/><jats:tex-math>$d\\ge 1$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. In this paper, we show that <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832300038X_inline6.png\"\/><jats:tex-math>$p_{\\textrm{c}}=1\/e+CL^{-d}+O(L^{-d-1})$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> for all <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832300038X_inline7.png\"\/><jats:tex-math>$d\\gt 8$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, where the model-dependent constant <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832300038X_inline8.png\"\/><jats:tex-math>$C$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> has the random-walk representation\n<jats:disp-formula><jats:alternatives><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" mimetype=\"image\" position=\"float\" xlink:href=\"S096354832300038X_eqnU1.png\"\/><jats:tex-math>\\begin{align*} C_{\\textrm{LT}}=\\sum _{n=2}^\\infty \\frac{n+1}{2e}U^{*n}(o),&amp;&amp; C_{\\textrm{LA}}=C_{\\textrm{LT}}-\\frac 1{2e^2}\\sum _{n=3}^\\infty U^{*n}(o), \\end{align*}<\/jats:tex-math><\/jats:alternatives><\/jats:disp-formula>where <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832300038X_inline9.png\"\/><jats:tex-math>$U^{*n}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is the <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832300038X_inline10.png\"\/><jats:tex-math>$n$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-fold convolution of the uniform distribution on the <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832300038X_inline11.png\"\/><jats:tex-math>$d$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-dimensional ball <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832300038X_inline12.png\"\/><jats:tex-math>$\\{x\\in{\\mathbb R}^d\\;: \\|x\\|\\le 1\\}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. The proof is based on a novel use of the lace expansion for the 2-point function and detailed analysis of the 1-point function at a certain value of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832300038X_inline13.png\"\/><jats:tex-math>$p$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> that is designed to make the analysis extremely simple.<\/jats:p>","DOI":"10.1017\/s096354832300038x","type":"journal-article","created":{"date-parts":[[2023,11,20]],"date-time":"2023-11-20T07:14:29Z","timestamp":1700464469000},"page":"238-269","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":3,"title":["Spread-out limit of the critical points for lattice trees and lattice animals in dimensions"],"prefix":"10.1017","volume":"33","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2273-4497","authenticated-orcid":false,"given":"Noe","family":"Kawamoto","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0943-7842","authenticated-orcid":false,"given":"Akira","family":"Sakai","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2023,11,20]]},"reference":[{"key":"S096354832300038X_ref4","doi-asserted-by":"publisher","DOI":"10.1214\/13-AOP843"},{"key":"S096354832300038X_ref21","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548313000102"},{"key":"S096354832300038X_ref26","doi-asserted-by":"publisher","DOI":"10.1007\/s00220-014-2256-x"},{"key":"S096354832300038X_ref7","doi-asserted-by":"publisher","DOI":"10.1007\/BF02108785"},{"key":"S096354832300038X_ref25","doi-asserted-by":"publisher","DOI":"10.1007\/s00220-007-0227-1"},{"key":"S096354832300038X_ref27","doi-asserted-by":"publisher","DOI":"10.1007\/s00220-022-04354-5"},{"key":"S096354832300038X_ref2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01206182"},{"key":"S096354832300038X_ref1","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.22021"},{"key":"S096354832300038X_ref14","doi-asserted-by":"publisher","DOI":"10.1007\/s004400100175"},{"key":"S096354832300038X_ref18","volume-title":"The Self-Avoiding Walk","author":"Madras","year":"1993"},{"key":"S096354832300038X_ref24","doi-asserted-by":"publisher","DOI":"10.1023\/A:1010320523031"},{"key":"S096354832300038X_ref3","doi-asserted-by":"publisher","DOI":"10.1007\/s00440-007-0101-2"},{"key":"S096354832300038X_ref9","doi-asserted-by":"publisher","DOI":"10.1007\/BF02099530"},{"key":"S096354832300038X_ref10","doi-asserted-by":"publisher","DOI":"10.1007\/BF01049008"},{"key":"S096354832300038X_ref13","doi-asserted-by":"publisher","DOI":"10.1214\/EJP.v15-783"},{"key":"S096354832300038X_ref16","doi-asserted-by":"publisher","DOI":"10.1063\/1.441869"},{"key":"S096354832300038X_ref19","unstructured":"[19] Miranda, Y. M. (2012) The critical points of lattice trees and lattice animals in high dimensions. Ph.D thesis, University of British Columbia."},{"key":"S096354832300038X_ref6","doi-asserted-by":"publisher","DOI":"10.1214\/aop\/1046294314"},{"key":"S096354832300038X_ref22","doi-asserted-by":"publisher","DOI":"10.1214\/aop\/1176989001"},{"key":"S096354832300038X_ref20","first-page":"129","article-title":"The growth constants of lattice trees and lattice animals in high dimensions","volume":"16","author":"Miranda","year":"2011","journal-title":"Electron. Commun. Probab."},{"key":"S096354832300038X_ref5","doi-asserted-by":"publisher","DOI":"10.1007\/s10955-021-02816-z"},{"key":"S096354832300038X_ref8","doi-asserted-by":"publisher","DOI":"10.1007\/BF01334760"},{"key":"S096354832300038X_ref12","doi-asserted-by":"publisher","DOI":"10.1007\/s00440-004-0405-4"},{"key":"S096354832300038X_ref11","doi-asserted-by":"publisher","DOI":"10.1214\/EJP.v9-224"},{"key":"S096354832300038X_ref15","doi-asserted-by":"publisher","DOI":"10.4153\/CJM-1967-080-4"},{"key":"S096354832300038X_ref17","unstructured":"[17] Liang, Y. (2022) Critical point for spread-out lattice trees in dimensions $d\\gt 8$ . Master thesis, Hokkaido University."},{"key":"S096354832300038X_ref23","doi-asserted-by":"publisher","DOI":"10.1007\/BF02186829"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S096354832300038X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,2,5]],"date-time":"2024-02-05T11:22:59Z","timestamp":1707132179000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S096354832300038X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,11,20]]},"references-count":27,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2024,3]]}},"alternative-id":["S096354832300038X"],"URL":"https:\/\/doi.org\/10.1017\/s096354832300038x","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,11,20]]},"assertion":[{"value":"\u00a9 The Author(s), 2023. 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