Mini-course: Multi-critical Schur measures and unitary matrix models.

Dan Betea (KU Leuven)

15-Jan-2021, 09:30-11:00 (3 years ago)

Abstract: We start by reviewing classical equalities between certain multiplicative Haar expectations over the unitary group (partition functions for certain classes of random unitary matrices), Toeplitz (and eventually Fredholm) determinants, and extremal/edge statistics of Okounkov's Schur measure. We pass by Heine's identity, the Gessel identity, the Borodin–Okounkov–Geronimo–Case identity, and Szego's strong theorem (if time permits). This brief tour aims to sketch the deep connections between random unitary matrices and symmetric functions. Such connections were first observed by Diaconis–Shashahani and later put to great use by Johansson, Rains, and collaborators.

We then aim at proving a recent result of the author, joint with J. Bouttier and H. Walsh (arXiv'd here arxiv.org/abs/2012.01995), 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 classical Baik–Deift–Johansson theorem on longest increasing subsequences of random permutations. In passing, we try to mention some related results both old: limit shape results for the random partitions under consideration; the associated phase transitions of Gross–Witten and Johansson; the original approach to multi-criticality of Periwal–Shevitz; the Schrodinger approach of Le Doussal–Majumdar–Schehr; and some recent work of Cafasso–Claeys–Girotti and Krajenbrink.

statistical mechanicsmathematical physicsprobability

Audience: learners

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Séminaire MEGA

Series comments: Description: Monthly seminar on random matrices and random graphs

Organizers: Guillaume Barraquand*, Laure Dumaz
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