{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T07:46:14Z","timestamp":1777448774981,"version":"3.51.4"},"reference-count":0,"publisher":"SAGE Publications","issue":"3-4","license":[{"start":{"date-parts":[[2009,10,1]],"date-time":"2009-10-01T00:00:00Z","timestamp":1254355200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Asymptotic Analysis"],"published-print":{"date-parts":[[2009,10]]},"abstract":"<jats:p>\n                    In this paper, we study the asymptotics of polynomials orthogonal with respect to the varying quartic weight \u03c9(x)=e\n                    <jats:sup>\u2212nV(x)<\/jats:sup>\n                    , where V(x)=V\n                    <jats:sub>t<\/jats:sub>\n                    (x)=x\n                    <jats:sup>4<\/jats:sup>\n                    \/4+x\n                    <jats:sup>2<\/jats:sup>\n                    t\/2. We focus on the critical case t=\u22122, in the sense that for t\u2265\u22122, the support of the associated equilibrium measure is a single interval, while for t&lt;\u22122, the support consists of two intervals. Globally uniform asymptotic expansions are obtained for z in three unbounded regions. These regions together cover the whole complex z-plane. In particular, in the region containing the origin, the expansion involves the \u03a8 function affiliated with the Hastings\u2013McLeod solution of the second Painlev\u00e9 equation. Our approach is based on a modified version of the steepest-descent method for Riemann\u2013Hilbert problems introduced by Deift and Zhou (Ann. Math. 137 (1993), 295\u2013370).\n                  <\/jats:p>","DOI":"10.3233\/asy-2008-0937","type":"journal-article","created":{"date-parts":[[2019,11,29]],"date-time":"2019-11-29T19:07:12Z","timestamp":1575054432000},"page":"125-154","source":"Crossref","is-referenced-by-count":3,"title":["Global asymptotics for polynomials orthogonal with exponential quartic weight"],"prefix":"10.1177","volume":"64","author":[{"given":"R.","family":"Wong","sequence":"first","affiliation":[{"name":"Liu Bie Ju Centre for Mathematical Sciences, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"L.","family":"Zhang","sequence":"additional","affiliation":[{"name":"Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"179","published-online":{"date-parts":[[2009,10,1]]},"container-title":["Asymptotic Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/ASY-2008-0937","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/ASY-2008-0937","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T12:38:55Z","timestamp":1777379935000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.sagepub.com\/doi\/10.3233\/ASY-2008-0937"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,10]]},"references-count":0,"journal-issue":{"issue":"3-4","published-print":{"date-parts":[[2009,10]]}},"alternative-id":["10.3233\/ASY-2008-0937"],"URL":"https:\/\/doi.org\/10.3233\/asy-2008-0937","relation":{},"ISSN":["0921-7134","1875-8576"],"issn-type":[{"value":"0921-7134","type":"print"},{"value":"1875-8576","type":"electronic"}],"subject":[],"published":{"date-parts":[[2009,10]]}}}