{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,28]],"date-time":"2026-08-28T05:02:23Z","timestamp":1787893343389,"version":"build-2784847793"},"reference-count":0,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2002,2,14]],"date-time":"2002-02-14T00:00:00Z","timestamp":1013644800000},"content-version":"unspecified","delay-in-days":44,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2002,1]]},"abstract":"<jats:p>We consider the problem of sampling \u2018unlabelled structures\u2019, <jats:italic>i.e.<\/jats:italic>, sampling combinatorial \nstructures modulo a group of symmetries. The main tool which has been used for this \nsampling problem is Burnside\u2019s lemma. In situations where a significant proportion of the \nstructures have no nontrivial symmetries, it is already fairly well understood how to apply \nthis tool. More generally, it is possible to obtain nearly uniform samples by simulating \na Markov chain that we call the Burnside process: this is a random walk on a bipartite \ngraph which essentially implements Burnside\u2019s lemma. For this approach to be feasible, \nthe Markov chain ought to be \u2018rapidly mixing\u2019, <jats:italic>i.e.<\/jats:italic>, converge rapidly to equilibrium. The \nBurnside process was known to be rapidly mixing for some special groups, and it has even \nbeen implemented in some computational group theory algorithms. In this paper, we show \nthat the Burnside process is not rapidly mixing in general. In particular, we construct an \ninfinite family of permutation groups for which we show that the mixing time is exponential \nin the degree of the group.<\/jats:p>","DOI":"10.1017\/s096354830100493x","type":"journal-article","created":{"date-parts":[[2003,10,15]],"date-time":"2003-10-15T15:59:58Z","timestamp":1066233598000},"page":"21-34","source":"Crossref","is-referenced-by-count":10,"title":["The \u2018Burnside Process\u2019 Converges Slowly"],"prefix":"10.1017","volume":"11","author":[{"given":"LESLIE ANN","family":"GOLDBERG","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"MARK","family":"JERRUM","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2002,2,14]]},"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S096354830100493X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,4]],"date-time":"2019-04-04T19:19:35Z","timestamp":1554405575000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S096354830100493X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2002,1]]},"references-count":0,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2002,1]]}},"alternative-id":["S096354830100493X"],"URL":"https:\/\/doi.org\/10.1017\/s096354830100493x","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2002,1]]}}}