Schedule

Title: Gibbs measures on subshifts

Abstract: We consider some of the main notions of Gibbs measures on subshifts introduced by different communities, such as dynamical systems, probability, operator algebras, and mathematical physics. For potentials with d-summable variation, we prove that several of the definitions considered by these communities are equivalent. In particular, when the subshift is of finite type (SFT), we show that all definitions coincide. In addition, we introduced a groupoid approach to describe some Gibbs measures, allowing us to show the equivalence between Gibbs measures and KMS states (the quantum analog of the Gibbs measures).  

Title: Reverse data-processing inequalities for partial transposes of random quantum states

Abstract: The quantum data processing inequality (QDPI) states that quantum relative entropy cannot be increased by local physical operations. However, the partial transposition can be used to increase the quantum relative entropy on random quantum states with probability one. We explain why such an unexpected phenomenon happens in view of random matrix theory and free probability.

Title: Conservation operator processes and their CLT from asymptotic representation theory

Abstract: In this talk, we examine applications of the theory of operator-valued processes to algebraic methods in probability theory. In particular, we analyze the asymptotic behavior of conservation operator processes, which are defined on a symmetric Fock space, derived from unitary groups and quantum unitary groups as their rank tends to infinity. 

Title: Spectral analysis of q-deformed unitary ensembles with the Al-Salam-Carlitz weight

Abstract: Al-Salam–Carlitz polynomial is a family of basic hypergeometric orthogonal polynomials in the Askey Scheme, which arises in a natural generalization of q-deformed Gaussian unitary ensemble. In this talk, I will first introduce Flajolet and Viennot’s theory concerning the combinatorics of the spectral moments of orthogonal polynomials, which yields a sign-definite sum formula of spectral moments. This leads to an explicit description of the limiting spectral density. This talk is based on joint work with Sung-Soo Byun and Jaeseong Oh.

Title: What is the probability that almost all eigenvalues being real for a large asymmetric random matrix?

Abstract: The Ginibre Orthogonal Ensemble (GinOE) is a real asymmetric random matrix model whose entries are independent standard Gaussian variables. As the matrix dimension grows, the eigenvalues of the GinOE tend to uniformly occupy the unit disk on the complex plane. At the same time, a nontrivial accumulation of eigenvalues occurs on the real axis, a phenomenon often referred as the "Saturn effect". Much of the existing literature has focused on the statistics of the eigenvalue count along the real line. While its mean and fluctuations have been studied in detail, the full distribution remains largely unknown. In this talk, we focus on the rare event that almost all eigenvalues lies on the real line, except for finitely many outliers. We present precise asymptotics for the probability of this degeneration and provide potential theoretic characterization of the estimation. We extend analysis to the elliptic GinOE, a one-parameter generalization of the GinOE that interpolates between the GinOE and the Gaussian Orthogonal Ensemble by tuning the asymmetry parameter. This is based on joint work with Gernot Akemann and Sung-Soo Byun.

Title: The FKG inequality for contact process in an evolving random environment and its applications

Abstract: The contact process with dynamic edges (CPDE) is a model for the spread of infection in which the state of each edge transmitting infection, available or unavailable, switches randomly over time. It is well known that the FKG inequality holds for the time-homogeneous contact process. This inequality is one of the most important tools for analyzing the properties of the process. In this talk, I will prove that the FKG inequality also holds for the CPDE. In addition, based on this result, I will present a hyperscaling inequality and show that the critical CPDE dies out.

Title: Abstract harmonic analysis via Fourier algebras

Abstract: We will make a survey on abstract harmonic analysis topics with Fourier algebras on the main focus. The topic includes amenability and complexification of Lie groups.