[{"date_updated":"2026-10-19T07:39:17Z","das_tickbox":"0","oa_version":"Published Version","OA_place":"publisher","file_date_updated":"2026-10-19T07:33:58Z","mathsc":["05C80"],"researchdata_availability":"no","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","type":"journal_article","author":[{"last_name":"Hollom","first_name":"Lawrence","full_name":"Hollom, Lawrence"},{"first_name":"Lyuben","last_name":"Lichev","id":"9aa8388e-d003-11ee-8458-c4c1d7447977","full_name":"Lichev, Lyuben"},{"full_name":"Mond, Adva","first_name":"Adva","last_name":"Mond"},{"last_name":"Portier","first_name":"Julien","full_name":"Portier, Julien"},{"first_name":"Yiting","last_name":"Wang","full_name":"Wang, Yiting","id":"1917d194-076e-11ed-97cd-837255f88785"}],"title":"Monotonicity and decompositions of random regular graphs","article_processing_charge":"Yes (in subscription journal)","ddc":["500"],"publication":"Annals of Applied Probability","volume":36,"page":"4716-4738","article_type":"original","day":"01","citation":{"ieee":"L. Hollom, L. Lichev, A. Mond, J. Portier, and Y. Wang, “Monotonicity and decompositions of random regular graphs,” <i>Annals of Applied Probability</i>, vol. 36, no. 5. Institute of Mathematical Statistics, pp. 4716–4738, 2026.","chicago":"Hollom, Lawrence, Lyuben Lichev, Adva Mond, Julien Portier, and Yiting Wang. “Monotonicity and Decompositions of Random Regular Graphs.” <i>Annals of Applied Probability</i>. Institute of Mathematical Statistics, 2026. <a href=\"https://doi.org/10.1214/26-AAP2345\">https://doi.org/10.1214/26-AAP2345</a>.","short":"L. Hollom, L. Lichev, A. Mond, J. Portier, Y. Wang, Annals of Applied Probability 36 (2026) 4716–4738.","mla":"Hollom, Lawrence, et al. “Monotonicity and Decompositions of Random Regular Graphs.” <i>Annals of Applied Probability</i>, vol. 36, no. 5, Institute of Mathematical Statistics, 2026, pp. 4716–38, doi:<a href=\"https://doi.org/10.1214/26-AAP2345\">10.1214/26-AAP2345</a>.","ista":"Hollom L, Lichev L, Mond A, Portier J, Wang Y. 2026. Monotonicity and decompositions of random regular graphs. Annals of Applied Probability. 36(5), 4716–4738.","apa":"Hollom, L., Lichev, L., Mond, A., Portier, J., &#38; Wang, Y. (2026). Monotonicity and decompositions of random regular graphs. <i>Annals of Applied Probability</i>. Institute of Mathematical Statistics. <a href=\"https://doi.org/10.1214/26-AAP2345\">https://doi.org/10.1214/26-AAP2345</a>","ama":"Hollom L, Lichev L, Mond A, Portier J, Wang Y. Monotonicity and decompositions of random regular graphs. <i>Annals of Applied Probability</i>. 2026;36(5):4716-4738. doi:<a href=\"https://doi.org/10.1214/26-AAP2345\">10.1214/26-AAP2345</a>"},"project":[{"_id":"bd95085b-d553-11ed-ba76-e55d3349be45","grant_number":"101076777","name":"Randomness and structure in combinatorics"},{"_id":"e19fef4c-1f3d-11f0-ac04-b8b1af6886d8","name":"THRESHOLDS IN SEMI-RANDOM PROCESSES AND GRAPH EMBEDDINGS","grant_number":"E624"}],"doi":"10.1214/26-AAP2345","keyword":["Random regular graphs","coupling","contiguity","total variation distance","decomposition"],"quality_controlled":"1","oa":1,"intvolume":"        36","external_id":{"arxiv":["2505.22875"]},"file":[{"file_name":"2026_AnnalsAppliedProbability_Hollom.pdf","access_level":"open_access","file_id":"23165","creator":"dernst","date_created":"2026-10-19T07:33:58Z","success":1,"date_updated":"2026-10-19T07:33:58Z","content_type":"application/pdf","checksum":"46101e61d29cf40f0a204801d15a848f","file_size":320593,"relation":"main_file"}],"publisher":"Institute of Mathematical Statistics","has_accepted_license":"1","publication_status":"published","month":"10","_id":"23155","publication_identifier":{"eissn":["2168-8737"],"issn":["1050-5164"]},"year":"2026","date_created":"2026-10-18T22:01:52Z","OA_type":"hybrid","abstract":[{"text":"In this work we establish several monotonicity and decomposition results in the framework of random regular graphs. Among other results, we\r\nshow that, for a wide range of parameters d1 ≤ d2, there exists a coupling of\r\nG(n,d1) and G(n,d2) satisfying that G(n,d1) ⊆ G(n,d2) with high probability, confirming a conjecture of Gao, Isaev and McKay in a new regime.\r\nOur contributions include new tools for analysing contiguity and total variation distance between random regular graph models, a novel procedure for\r\ngenerating unions of random edge-disjoint perfect matchings, and refined estimates of Gao’s bounds on the number of perfect matchings in random regular graphs. In addition, we make progress towards another conjecture of Isaev,\r\nMcKay, Southwell and Zhukovskii.","lang":"eng"}],"status":"public","language":[{"iso":"eng"}],"department":[{"_id":"MaKw"},{"_id":"GradSch"}],"scopus_import":"1","acknowledgement":"We are grateful to Béla Bollobás, to Matthew Kwan and to Fabio\r\nToninelli for their helpful comments and suggestions, and to Nicholas Wormald for useful\r\nremarks on a preliminary version of this work and pointing us towards the reference [29].\r\nThe authors would also like to thank several anonymous referees for many useful suggestions,\r\nincluding pointing out that Lemma 2.10 already appeared in [11].\r\nPart of this research was done during a visit of the fourth author to IST Austria. We thank\r\nIST Austria for its hospitality.\r\nFor open access purposes, the authors have applied a CC BY public copyright license to\r\nany author accepted manuscript version arising from this submission. L. Hollom was supported by the Internal Graduate Studentship of Trinity College, Cambridge.\r\nL. Lichev was supported by the Austrian Science Fund (FWF) grant No. 10.55776/ESP624.\r\nA. Mond was supported by UK Research and Innovation grant MR/W007320/2. Wang was\r\nsupported by the ERC Starting Grant “RANDSTRUCT” No. 101076777.","date_published":"2026-10-01T00:00:00Z","supplementarymaterial":"no","arxiv":1,"fulldoi":"https://doi.org/10.1214/26-AAP2345","issue":"5","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"PlanS_conform":"1"},{"citation":{"ama":"Barton NH, Etheridge A, Sturm A. Coalescence in a Random Background. <i>Annals of Applied Probability</i>. 2004;14(2):754-785. doi:<a href=\"https://doi.org/10.1214/105051604000000099\">10.1214/105051604000000099</a>","apa":"Barton, N. H., Etheridge, A., &#38; Sturm, A. (2004). Coalescence in a Random Background. <i>Annals of Applied Probability</i>. Institute of Mathematical Statistics. <a href=\"https://doi.org/10.1214/105051604000000099\">https://doi.org/10.1214/105051604000000099</a>","ista":"Barton NH, Etheridge A, Sturm A. 2004. Coalescence in a Random Background. Annals of Applied Probability. 14(2), 754–785.","mla":"Barton, Nicholas H., et al. “Coalescence in a Random Background.” <i>Annals of Applied Probability</i>, vol. 14, no. 2, Institute of Mathematical Statistics, 2004, pp. 754–85, doi:<a href=\"https://doi.org/10.1214/105051604000000099\">10.1214/105051604000000099</a>.","chicago":"Barton, Nicholas H, Alison Etheridge, and Anja Sturm. “Coalescence in a Random Background.” <i>Annals of Applied Probability</i>. Institute of Mathematical Statistics, 2004. <a href=\"https://doi.org/10.1214/105051604000000099\">https://doi.org/10.1214/105051604000000099</a>.","short":"N.H. Barton, A. Etheridge, A. Sturm, Annals of Applied Probability 14 (2004) 754–785.","ieee":"N. H. Barton, A. Etheridge, and A. Sturm, “Coalescence in a Random Background,” <i>Annals of Applied Probability</i>, vol. 14, no. 2. Institute of Mathematical Statistics, pp. 754–785, 2004."},"page":"754 - 785","article_type":"original","extern":"1","day":"01","quality_controlled":"1","oa":1,"intvolume":"        14","main_file_link":[{"url":" https://doi.org/10.48550/arXiv.math/0406174","open_access":"1"}],"doi":"10.1214/105051604000000099","OA_place":"repository","oa_version":"Preprint","date_updated":"2026-05-29T09:59:37Z","publication":"Annals of Applied Probability","volume":14,"user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","type":"journal_article","title":"Coalescence in a Random Background","author":[{"first_name":"Nicholas H","last_name":"Barton","full_name":"Barton, Nicholas H","id":"4880FE40-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-8548-5240"},{"full_name":"Etheridge, Alison","first_name":"Alison","last_name":"Etheridge"},{"last_name":"Sturm","first_name":"Anja","full_name":"Sturm, Anja"}],"article_processing_charge":"No","scopus_import":"1","date_published":"2004-05-01T00:00:00Z","status":"public","publist_id":"1842","language":[{"iso":"eng"}],"fulldoi":"https://doi.org/10.1214/105051604000000099","issue":"2","arxiv":1,"month":"05","_id":"4253","external_id":{"arxiv":["0406174"]},"publisher":"Institute of Mathematical Statistics","publication_status":"published","year":"2004","date_created":"2018-12-11T12:07:52Z","OA_type":"green","abstract":[{"lang":"eng","text":"We consider a single genetic locus which carries two alleles, labelled P and Q. This locus experiences selection and mutation. It is linked to a second neutral locus with recombination rate r. If r = 0, this reduces to the study of a single selected locus. Assuming a Moran model for the population dynamics, we pass to a diffusion approximation and, assuming that the allele frequencies at the selected locus have reached stationarity, establish the joint generating function for the genealogy of a sample from the population and the frequency of the P allele. In essence this is the joint generating function for a coalescent and the random background in which it evolves. We use this to characterize, for the diffusion approximation, the probability of identity in state at the neutral locus of a sample of two individuals (whose type at the selected locus is known) as solutions to a system of ordinary differential equations. The only subtlety is to find the boundary conditions for this system. Finally, numerical examples are presented that illustrate the accuracy and predictions of the diffusion approximation. In particular, a comparison is made between this approach and one in which the frequencies at the selected locus are estimated by their value in the absence of fluctuations and a classical structured coalescent model is used."}],"publication_identifier":{"issn":["1050-5164"],"eissn":["2168-8737"]}}]