BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Dan Betea (KU Leuven) DTSTART:20210115T093000Z DTEND:20210115T110000Z DTSTAMP:20260423T022715Z UID:MEGA/12 DESCRIPTION:Title: Mi ni-course: Multi-critical Schur measures and unitary matrix models.\nb y Dan Betea (KU Leuven) as part of Séminaire MEGA\n\n\nAbstract\nWe start by reviewing classical equalities between certain multiplicative Haar exp ectations over the unitary group (partition functions for certain classes of random unitary matrices)\, Toeplitz (and eventually Fredholm) determina nts\, and extremal/edge statistics of Okounkov's Schur measure. We pass by Heine's identity\, the Gessel identity\, the Borodin–Okounkov–Geronim o–Case identity\, and Szego's strong theorem (if time permits). This bri ef tour aims to sketch the deep connections between random unitary matrice s and symmetric functions. Such connections were first observed by Diaconi s–Shashahani and later put to great use by Johansson\, Rains\, and colla borators.\n\nWe then aim at proving a recent result of the author\, joint with J. Bouttier and H. Walsh (arXiv'd here https://arxiv.org/abs/2012.019 95)\, which shows that when the unitary matrix model potential is tuned “multi-critically”\, all the quantities above tend to the higher-order Tracy–Widom distributions introduced recently by Le Doussal–Majumdar –Schehr. This result is a gap probability result for the largest part of the associated random partition\, and as such extends the by now classica l Baik–Deift–Johansson theorem on longest increasing subsequences of r andom permutations. In passing\, we try to mention some related results bo th old: limit shape results for the random partitions under consideration\ ; the associated phase transitions of Gross–Witten and Johansson\; the o riginal approach to multi-criticality of Periwal–Shevitz\; the Schroding er approach of Le Doussal–Majumdar–Schehr\; and some recent work of Ca fasso–Claeys–Girotti and Krajenbrink.\n LOCATION:https://researchseminars.org/talk/MEGA/12/ END:VEVENT END:VCALENDAR