Organizers：Jiaming Li & Dong Yao
Time：March 29, 2025
Venue：E4-233
8:40-9:20
Host：Jiaming Li, AMSS
Speaker：Dangzheng Liu, University of Science and Technology of China
Title：Method of polynomial moments in Random Matrix Theory
Abstract：The method of moments, a cornerstone technique in probability theory, was first rigorously employed by Pafnuty Chebyshev in 1887 to establish a proof of the Central Limit Theorem. Its significance expanded into random matrix theory by Eugene Wigner in 1955 to derive the celebrated Wigner semicircle law, while Alexander Soshnikov leveraged it in 1999 to characterize the Tracy-Widom distribution for Wigner matrices. Notably, it continues to demonstrate formidable power with RMT, from random band matrices to inhomogeneous ensembles. This talk provides a concise survey.
9:20-10:00
Host：Jiaming Li, AMSS
Speaker：Meng Yang, Great Bay University
Title：2D Coulomb Gases and Partition Functions
Abstract：We consider 2D Coulomb gases with the external potential Q(z)=|z|^2-2c log|z-a|, where c>0 and a \in C. Equivalently, this model can be realised as N eigenvalues of the complex Ginibre matrix of size (c+1) N(c+1) N conditioned to have deterministic eigenvalue a with multiplicity cN. Depending on the values of c and a, the droplet reveals a phase transition: it is doubly connected in the post-critical regime and simply connected in the pre-critical regime. In both regimes, we derive precise large-N expansions of the free energy up to the O(1) term, providing a non-radially symmetric example that confirms the Zabrodin-Wiegmann conjecture made for general planar Coulomb gas ensembles. As a consequence, our results provide asymptotic behaviours of moments of the characteristic polynomial of the complex Ginibre matrix, where the powers are of order O(N). Furthermore, by combining with a duality formula, we obtain precise large deviation probabilities of the smallest eigenvalue of the Laguerre unitary ensemble. This talk is based on the joint work with Sung-Soo Byun and Seong-Mi Seo.
10:20-11:00
Host：Zhonggen Su, Zhejiang University
Speaker：Zhigang Bao, University of Hong Kong
Title：Decorrelation transition for Wigner minor process
Abstract：In the past decade or so, significant progress has been made in research on the universality problem of random matrix spectral statistics. While most of the work has focused on the spectral statistics of a single random matrix, some understanding has also been gained regarding the relationship between the spectral statistics of two dependent or coupled matrices. In this talk, we will discuss a model that is well understood in the Gaussian setting but remains less explored in the general setting: the Wigner minor process. Specifically, we will demonstrate that a correlation-decorrelation transition, originally known for the GUE minor process, also holds for the Wigner minor process.
This is based on a joint work with Giorgio Cipolloni, Laszlo Erdos, Joscha Henheik and Oleksii Kolupaiev.
11:00-11:40
Host：Zhonggen Su, Zhejiang University
Speaker：Qiang Zeng, AMSS
Title：Hessian spectrum at the global minimum of elastic manifold
Abstract：In statistical physics, elastic manifold is a model for disordered elastic systems introduced by Fisher (1986), Mezard and Parisi (1991). In 2020, Fyodorov and Le Doussal predicted phase transitions for the Hessian spectrum of elastic manifold at the ground state using Parisi's replica trick. In this talk, we will show that their predictions hold in the so-called replica symmetric regime based on counting critical points of the Hamiltonian via the Kac–Rice formula. Our tools include the matrix Dyson equation and free convolution, among others.
13:30-14:10
Host：Zhigang Bao, University of Hong Kong
Speaker：Shihao Li, Sichuan University
Title：q-正交系综的关联核及其应用
Abstract：近年来，离散系综在随机组合模型中扮演了非常重要的角色。在本报告中，我们将讨论定义在指数格子上的正交系综（即q-正交系综），以及对应的关联核函数。利用q-正交多项式，我们定义了q-斜正交多项式，并说明了q-正交系综的关联核函数可以表示成q-酉系综的秩一摄动。
14:10-14:50
Host：Zhigang Bao, University of Hong Kong
Speaker：Dong Wang, University of Chinese Academy of Sciences
Title：Local statistics of Muttalib-Borodin ensemble in the hard to soft transitive regime and the related integrability
Abstract：Muttalib-Borodin ensemble is a typical biorthogonal ensemble. It has a hard edge limit that is expressed by Wright's generalized Bessel functions, or by Meijer G functions if the parameter $\theta$ is an integer. This hard edge limit does not occur in orthogonal polynomial ensembles unless $\theta = 1$. In this talk we consider the transition from the hard edge limit to the soft edge limit of the Muttalib-Borodin ensemble with an integer $\theta$ parameter, and show that the limiting distributions are related to Painleve-type equations. Our result generalizes the known relation between the transitive regime of orthogonal polynomial ensemble and the Painleve XXXIV. This is joint work with Shui-Xia Xu.
15:10-15:50
Host：Dong Yao, Jiangsu Normal University
Speaker：Wangjun Yuan, Southern University of Science and Technology
Title：Strong convergence for tensor GUE random matrices
Abstract：Haagerup and Thorbj{\o}rnsen proved that iid GUEs converge strongly to free semicircular elements as the dimension grows to infinity. Motivated by considerations from quantum physics -- in particular, understanding nearest neighbor interactions in quantum spin systems -- we consider iid GUE acting on multipartite state spaces, with components on more than half of sites and identity on the remaining sites. In particular, the GUEs have mixing components on some sites. We show that under proper assumptions on the dimension of the sites, strong asymptotic freeness still holds. Our proof relies on an interpolation technology recently introduced by Bandeira, Boedihardjo and van Handel.
This is a joint work with Benoit Collins.
16:00-17:30
Free Discussion
