## 2023 Fall School in Probability I

**活动时间：** 从 2023-10-08 09:30 到 2023-10-13 17:00

**场地：** 北京国际数学研究中心，镜春园82号甲乙丙楼报告厅

**Venue:** **Research courses:** Jia Yi
Bing Lecture Hall, Beijing International Center for Mathematical Research

** Pao-Lu Hsu Lecture:** Si
Yuan Hall, Zhi Hua Building, School of Mathematical Sciences

The 2023 Fall School in Probability I is part of the PKU Thematic Semester Program on Stochastics to celebrate the 110th anniversary of Mathematical Sciences at Peking University. In parellel with the Fall School, Qi-Man Shao (SUSTech) will also deliver this year’s Pao-Lu Hsu Lecture (hosted by the School of Mathematical Sciences) on 13 October.

**Research course lecturers and titles:**

**Amir Dembo (Stanford):** Large deviations, low
complexity Gibbs measures and exponential random graph models

**Martin Hairer (EPFL & Imperial):** Stochastic Yang-Mills

**Xin Sun (BICMR): **Liouville conformal field theory and random geometry

**Pao-Lu Hsu Lecturer and
title:**

**Qi-Man Shao
(SUSTech):** From Lindeberg’s central limit theorem to the Cramér moderate deviation theorem

(tips: The first class by Dembo meets on Wednesday 9:00am instead of 9:30.)

**Abstracts: **

**Amir Dembo: **

I will overview the emerging theory of large deviations for low complexity Gibbs measures, the naive mean field approximation of their partition functions

and representing such measures as mixtures of not too many product measures. We will consider certain applications, such as to the abundance of specific patterns in sparse random graphs, having many arithmetic progressions in a uniformly chosen random set and the universality of the Potts model on graphs of growing average degrees. In particular, we shall see its sharp conclusions about typical samples from Exponential Random Graph Models, that are widely used in the analysis of social networks.

**Xin Sun: **

Liouville quantum gravity (LQG) is a theory of random surfaces that describes the scaling limit of natural discrete random surfaces such as uniform triangulations. The geometry of the random surfaces is governed by a variant of Gaussian free field called Liouville conformal field theory (CFT). This is an important example of 2D CFT, which was rigorously and exactly solved in recent years. In the first lecture I will explain the relation between Liouville CFT and random surfaces. An important aspect of LQG is its coupling with 2D critical systems that can be described by Schramm-Loewner evolution (SLE) in the continuum. In the second lecture I will demonstrate how the exact solvability of Liouville CFT combined with the SLE/LQG coupling can give new exact results for SLE, some of which confirms predictions in physics while others are new to physicists.

Participants are expected to arrange their own meals and accommodation. Participants from outside Peking University who need us to arrange their access to the PKU campus should send an email to probability@math.pku.edu.cn by 11am Wednesday, 27 September with the following information: name, title, affiliation, national ID number (or passport if foreign citizen) and phone number. PhD students and postdocs should also include in the email the name of their supervisor and a paragraph describing their research. The subject of the email should be “Fall School in Probability I: campus access”.

Since campus access for groups needs to be arranged in advance and 1--7 October is the national holiday, we may not be able to deal with requests received after the above deadline.