Date: August 28 to September 1, 2017

Main Lecturer: Alice Guionnet (Lyon)

Location: Columbia Mathematics Department

Funding: NSF CBMS conference grant DMS-1642595 and the Minerva Lecture Series

Organizers: Ivan Corwin and Yi Sun

Conference Poster: Poster

Understanding the large dimension asymptotics of random matrices or related models such as random tilings has been a hot topic for the last twenty years within probability, mathematical physics, and statistical mechanics. Because such models are highly correlated, classical methods based on independent variables fail. Special cases have been studied in detail thanks to specific forms of the laws, such as determinantal laws. These lectures will investigate a general class of models using the so called Dyson-Schwinger equations or generalizations such as Nekrasov's equations. The idea is similar to Stein's method in that the observables are approximate solutions of equations that can be solved asymptotically.

Alice Guionnet (Lyon) will give ten main lectures, divided into two per day. Besides these lectures there will be supplementary lectures by other senior researchers attending the school, including:

- Gaetan Borot (Bonn)

**Title:**1D discrete log gas in the macroscopic regime

**Abstract:**We will explain how fluctuations of linear statistics and asymptotic expansion of the partition function and correlation function of a 1d discrete log-gas can be carried out, in full analogy with the (continuous) 1d log gas/beta-ensembles. This discrete model is realized in 2d random lozenge tilings. In particular, we will describe the replacement for Selberg integral (with Jack binomial model) and of Dyson-Schwinger equations (with Nekrasov equations) which are instrumental in studying this discrete model. The talk is based on joint ongoing work with Gorin and Guionnet. - Paul Bourgade (NYU)
- Vadim Gorin (MIT)
- Sylvia Serfaty (NYU)

Lecture | Description |
---|---|

Lectures 1 + 2 | Law of large numbers in random matrices (light / heavy tailed entries), and concentration of measure. |

Lectures 3 + 4 | Central limit theorem in the one cut case (light / heavy tailed entries). |

Lectures 5 + 6 | Generalization to the discrete setting of tiling models. |

Lectures 7 + 8 | Topological expansions for large N asymptotics of partition functions. |

Lectures 9 + 10 | Generalization to several cut models. |

Time | Description | Location |
---|---|---|

09:30 am - 10:00 am | Breakfast | Room 508 |

10:00 am - 11:00 am | AM Lecture (Guionnet) | Room 312 |

11:00 am - 11:30 am | Break | Room 508 |

11:30 am - 12:30 pm | Participant Talks | Room 312 |

12:30 pm - 02:00 pm | Lunch | Room 508 |

02:00 pm - 03:00 pm | PM Lecture (Guionnet) | Room 312 |

03:00 pm - 04:30 pm | Problem Session | Room 312 |

04:30 pm - 05:00 pm | Break | Room 508 |

05:00 pm - 06:00 pm | Supplemental Lecture / Additional Discussion | Room 312 |

06:00 pm - 07:00 pm | Dinner | Room 508 |