{"id":237,"date":"2023-01-13T14:05:13","date_gmt":"2023-01-13T14:05:13","guid":{"rendered":"https:\/\/do.csc.knu.ua\/?p=237"},"modified":"2025-12-22T10:02:58","modified_gmt":"2025-12-22T10:02:58","slug":"lah-distributions-stirling-numbers-and-phase-transitions-for-convex-hulls-of-random-walks","status":"publish","type":"post","link":"https:\/\/do.csc.knu.ua\/?p=237","title":{"rendered":"Lah distributions, Stirling numbers, and phase transitions for convex hulls of random walks"},"content":{"rendered":"<p class=\"contentheading\"><span style=\"font-size: 1.94em;\">\\(\\)<\/span><span style=\"font-size: 1.94em;\">21 \u043a\u0432\u0456\u0442\u043d\u044f 2021 \u0440\u043e\u043a\u0443 \u043e 14 \u0433\u043e\u0434 15 \u0445\u0432<\/span><\/p>\n<div class=\"article-content\">\n<p>Let \\(X_1,X_2,\\ldots\\) be independent random vectors in \\(\\mathbb{R}^d\\) having an absolutely continuous distribution. Consider the random walk \\(S_k:=X_1+\\ldots+X_k\\), and let \\(P_n:={\\rm conv}\\{0,S_1,S_2,\\ldots,S_n\\}\\) be the convex hull of its first \\(n\\) steps. We shall be interested in the number of the \\(k\\)-dimensional faces of the polytope \\(P_n\\) and in particular, whether this number is equal to the maximal possible number \\({n+1\\choose k+1}\\) with high probability, as \\(n\\), \\(d\\), and possibly also \\(k\\) diverge to \\(\\infty\\). There is an explicit formula for the expected number of \\(k\\)-dimensional faces which involves Stirling numbers of both kinds. Motivated by this formula, we introduce a distribution, called the Lah distribution, and study its properties.<\/p>\n<p class=\"western\" lang=\"uk-UA\" align=\"justify\">\u0414\u043e\u043f\u043e\u0432\u0456\u0434\u0430\u0447:\u00a0<strong>Zakhar Kabluchko (University of M\u00fcnster, Germany)<\/strong><\/p>\n<p>\u0414\u0430\u0442\u0430 \u043f\u0440\u043e\u0432\u0435\u0434\u0435\u043d\u043d\u044f: 21 \u043a\u0432\u0456\u0442\u043d\u044f 2021 \u0440\u043e\u043a\u0443 \u043e 14 \u0433\u043e\u0434 15 \u0445\u0432.<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>21 \u043a\u0432\u0456\u0442\u043d\u044f 2021 \u0440\u043e\u043a\u0443 \u043e 14 \u0433\u043e\u0434 15 \u0445\u0432 Let \\(X_1,X_2,\\ldots\\) be independent random vectors in \\(\\mathbb{R}^d\\) having an absolutely continuous distribution. Consider the random walk \\(S_k:=X_1+\\ldots+X_k\\), and let \\(P_n:={\\rm conv}\\{0,S_1,S_2,\\ldots,S_n\\}\\) be the convex hull of its first \\(n\\) steps. We shall be interested in the number of the \\(k\\)-dimensional faces of the polytope \\(P_n\\)&#8230;<\/p>\n<p class=\"more-link-wrap\"><a href=\"https:\/\/do.csc.knu.ua\/?p=237\" class=\"more-link\">Read More<span class=\"screen-reader-text\"> &ldquo;Lah distributions, Stirling numbers, and phase transitions for convex hulls of random walks&rdquo;<\/span> &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[9],"tags":[],"class_list":["post-237","post","type-post","status-publish","format-standard","hentry","category-seminar"],"_links":{"self":[{"href":"https:\/\/do.csc.knu.ua\/index.php?rest_route=\/wp\/v2\/posts\/237","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/do.csc.knu.ua\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/do.csc.knu.ua\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/do.csc.knu.ua\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/do.csc.knu.ua\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=237"}],"version-history":[{"count":1,"href":"https:\/\/do.csc.knu.ua\/index.php?rest_route=\/wp\/v2\/posts\/237\/revisions"}],"predecessor-version":[{"id":1823,"href":"https:\/\/do.csc.knu.ua\/index.php?rest_route=\/wp\/v2\/posts\/237\/revisions\/1823"}],"wp:attachment":[{"href":"https:\/\/do.csc.knu.ua\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=237"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/do.csc.knu.ua\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=237"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/do.csc.knu.ua\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=237"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}