{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,1]],"date-time":"2026-05-01T00:50:54Z","timestamp":1777596654579,"version":"3.51.4"},"reference-count":17,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2021,1,4]],"date-time":"2021-01-04T00:00:00Z","timestamp":1609718400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Beijing Academy of Artificial Intelligence(BAAI)","award":["N.A."],"award-info":[{"award-number":["N.A."]}]},{"DOI":"10.13039\/501100002858","name":"China Postdoctoral Science Foundation","doi-asserted-by":"publisher","award":["Nos. 2020TQ0011"],"award-info":[{"award-number":["Nos. 2020TQ0011"]}],"id":[{"id":"10.13039\/501100002858","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this paper, we study the evolution of a Finitary Random Interlacement (FRI) with respect to the expected length of each fiber. In contrast to the previously proved phase transition between sufficiently large and small fiber length, for all d\u22653, FRI is NOT stochastically monotone as fiber length increases. At the same time, numerical evidence still strongly supports the existence and uniqueness of a critical fiber length, which is estimated theoretically and numerically to be an inversely proportional function with respect to system intensity.<\/jats:p>","DOI":"10.3390\/e23010069","type":"journal-article","created":{"date-parts":[[2021,1,4]],"date-time":"2021-01-04T08:35:19Z","timestamp":1609749319000},"page":"69","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["On (Non-)Monotonicity and Phase Diagram of Finitary Random Interlacement"],"prefix":"10.3390","volume":"23","author":[{"given":"Zhenhao","family":"Cai","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, Peking University, Beijing 100871, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yunfeng","family":"Xiong","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Peking University, Beijing 100871, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8408-5755","authenticated-orcid":false,"given":"Yuan","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Peking University, Beijing 100871, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,1,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"659","DOI":"10.1007\/s00039-019-00494-4","article-title":"Finitary random interlacements and the Gaboriau\u2013Lyons problem","volume":"29","author":"Bowen","year":"2019","journal-title":"Geom. Funct. Anal."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"2039","DOI":"10.4007\/annals.2010.171.2039","article-title":"Vacant Set of Random Interlacements and Percolation","volume":"171","author":"Sznitman","year":"2009","journal-title":"Ann. Math."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Sznitman, A.S. (2012). Topics in Occupation Times and Gaussian Free Fields, European Mathematical Society.","DOI":"10.4171\/109"},{"key":"ref_4","first-page":"1604","article-title":"Interlacement percolation on transient weighted graphs","volume":"14","author":"Teixeira","year":"2009","journal-title":"Electr. J. Probab."},{"key":"ref_5","unstructured":"Procaccia, E.B., Ye, J., and Zhang, Y. (2019). Percolation for the Finitary Random interlacements. arXiv."},{"key":"ref_6","unstructured":"Cai, Z., Han, X., Ye, J., and Zhang, Y. (2020). 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Math."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"692","DOI":"10.1214\/aop\/1176993514","article-title":"The binary contact path process","volume":"11","author":"Griffeath","year":"1983","journal-title":"Ann. Probab."},{"key":"ref_16","first-page":"289","article-title":"Uniqueness and non-uniqueness in percolation theory","volume":"3","author":"Jonasson","year":"2006","journal-title":"Probab. Surv."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Lyons, R., and Schramm, O. (2011). Indistinguishability of percolation clusters. Selected Works of Oded Schramm, Springer.","DOI":"10.1007\/978-1-4419-9675-6_23"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/1\/69\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:06:45Z","timestamp":1760159205000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/1\/69"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1,4]]},"references-count":17,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2021,1]]}},"alternative-id":["e23010069"],"URL":"https:\/\/doi.org\/10.3390\/e23010069","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,1,4]]}}}