Title & Abstract

Minicourses

Seonwoo Kim:  Resolvent approach to metastability: theory, applications, and extensions

Abstract.  In this three-lecture minicourse, I will introduce the recently developed resolvent approach [Landim-Marcondes-Seo, JEMS '25] to metastability. First, I will review the historical developments in the treatment of the metastability phenomenon and summarize the pros/cons of the previous methods. Next, I will state and prove the resolvent approach in a simple setting, focusing especially on its strength, along with some real model applications. Finally, I will summarize my recent work with Jungkyoung Lee (KIAS) to extend and generalize this approach to non-compact metastable transitions.

Junhee Ryu:  A Tutorial on Stochastic Partial Differential Equations (SPDEs)

Abstract.  In this minicourse, I will introduce the basic theories of stochastic partial differential equations (SPDEs). I will begin by introducing the motivations behind various equations, including those arising from filtering problems and superdiffusions. Then I will present the Sobolev regularity theory for SPDEs. Finally, I will highlight several topics that have been actively studied in recent years.

Invited Talks

Soobin Cho:  Approximate factorizations for non-symmetric jump processes

Abstract.  In this talk, we first discuss approximate factorizations of heat kernels and Green functions for purely discontinuous Markov processes. Under natural conditions, we show that the approximate factorization of the heat kernel is equivalent to that of the Green function. In the second part, we will present applications of these factorizations to derive two-sided heat kernel estimates for three classes of processes: stable-like processes with critical killing in $C^{1,Dini}$ open sets; killed stable-like processes with low regularity coefficients; and non-symmetric stable processes in $C^{1,2-Dini}$ open sets. In particular, we obtain sharp, explicit two-sided estimates for the killed and censored stable processes in $C^{1,Dini}$ open sets. This is based on joint work with Professor Renming Song (UIUC).

Hong Chang Ji:  Spectral estimators in general linear models

Abstract.  General linear model concerns the statistical problem of estimating a vector x from the vector of measurements y=Ax+e, where A is a given design matrix whose rows correspond to individual measurements and e represents errors in measurements. Popular iterative algorithms, e.g. message passing, used in this context requires a "warm start", meaning they must be initialized better than a random guess. In practice, it is often the case that a spectral estimator, i.e. a principal component of certain matrix built from Y, serves as such an initialization. In this talk, we discuss the theoretical aspect of the spectral estimator and present a theorem on its performance guarantee. The theorem gives a threshold for the sample complexity, that is, how many measurements are needed for a warm start to exist.

Jaehun Lee (KAIST):  Stability of heat kernel estimates for symmetric jump processes on metric measure spaces 

Abstract.  In this talk, we study symmetric pure jump Markov processes on metric measure spaces under volume doubling conditions. Our focus is on the stability of heat kernel estimates when the jumping kernel exhibits mixed polynomial growth. Unlike classical settings, the growth rate of the jumping kernel may not align with the scale function that determines near-and off-diagonal behaviors of the heat kernel. Assuming the lower scaling index of the scale function exceeds one, we establish the stability of heat kernel estimates under general conditions.

This talk is based on the joint work with Joohak Bae, Jaehoon Kang and Panki Kim.

Jaehun Lee (HKU):  Bounded rank perturbations of i.i.d. random matrices under minimal moment assumptions

Abstract.  In this talk, we investigate bounded rank perturbations of independent and identically distributed (i.i.d.) random matrices. Previously, it was established that spectral outliers of such perturbed matrices converge to the outlying eigenvalues of the perturbation matrix under the assumption of finite fourth moments. Recent advances have relaxed this condition to finite second moments, albeit with restrictive conditions on the perturbation matrix. We extend these results by demonstrating convergence under finite second moment assumptions for general bounded rank perturbations without additional constraints. Our approach is transparent and employs fundamental algebraic identities involving determinants. Additionally, we derive analogous results for sparse random matrix models. If time permits, we will discuss applications of these findings to signal detection problems. This talk is based on joint work with Zhigang Bao.

DongJun Min:  Generalized Analysis of Stochastic Dynamical Systems Perturbed by Heavy-Tailed Noise

Abstract.  Stochastic dynamical systems driven by heavy-tailed noise play a fundamental role in understanding complex behaviors in machine learning, optimization, and physical systems. Prior studies have examined escape phenomena and stability in such systems under structural assumptions on the dynamics—most notably, where the trajectory’s distance to a fixed boundary grows monotonically over time. These assumptions, while analytically convenient, limit the scope of applicability to specific system classes.

In this talk, we present a generalized analysis of stochastic dynamical systems perturbed by heavy-tailed noise without relying on monotonic growth conditions. By relaxing the structural assumptions on the deterministic flow, we uncover how the tail properties of the noise influence the stability transitions, and escape mechanisms even in non-monotonic or fluctuating dynamical regimes. This broader framework unifies and extends existing results, offering a more flexible and robust theoretical foundation for stochastic systems operating under minimal regularity.

Jiwoon Park:  Critical phi4 Model at and Above Dimension Four and Its Finite-Size Scaling

Abstract.  It is well known that the critical phi4 model in dimension four and higher exhibits mean-field behavior. This can be understood through the renormalisation group (RG), under which the effective potential converges to a quadratic form. Consequently, in the infrared limit, spin fields scale like those of the Gaussian free field. Although a rigorous justification of this behavior is challenging, the mechanism is universal, making models at the upper critical dimension particularly compelling.

However, finite-size effects suggest that mean-field behavior and the Gaussian scaling limit may not be consistent in finite volumes. This apparent discrepancy is resolved by the ensemble scaling limit, which interpolates between the two regimes. In this talk, I will explain how this interpolation arises naturally within the RG framework.

Kihoon Seong:  Central limit theorem for Φ^4 QFT measures in low temperature and thermodynamic limit

Abstract.  I will introduce basic concepts such as the concentration and fluctuation of Φ^4 Gibbs type measures from the perspectives of statistical physics, quantum field theory, probability theory, and PDEs. The focus will be on the low temperature behavior and the thermodynamic limit of these probability measures, with particular attention to fluctuations around the soliton manifold.

Jaeyun Yi:  Large-scale analysis of stochastic PDEs

Abstract.  In this talk, I will introduce how stochastic PDEs arise in complex systems such as statistical mechanics, fluid dynamics, and quantum field theory.  Unlike their deterministic counterparts, stochastic PDEs can exhibit rich solution behaviors at large scales, including multifractality and turbulent chaos. We will explore these phenomena through specific examples, highlighting the mathematical challenges and physical insights that SPDEs offer.

Contributed Talks (6/23 Mon)

Yeonggwang Jung:  Spectral Analysis of $q$-Al-Salam-Carlitz Unitary Ensembles

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.

Seungjoon Oh:  Spectral moments of non-Hermitian complex and symplectic random matrices

Abstract.  The spectral moments have been considered as fundamental objects in the history of random matrix theory, having been extensively studied mostly for Hermitian ensembles. Unlike the case of Hermitian random matrices where all eigenvalues are real, eigenvalues of non-Hermitian random matrices lie on the complex plane. In this talk, we will discuss the mixed spectral moments, which involves both holomorphic and anit-holomorphic parts, of a certain class of non-Hermitian random matrices. This includes the exactly solvable ensembles; the elliptic Ginibre ensemble, the non-Hermitian Wishart ensemble, and the planar Gegenbauer ensemble. We present a unifying and systematic framework for analysing mixed spectral moments and derive the explicit formula. We also investigate the large $N$-limit of the spectral moments, where $N$ is the dimension of the matrix. This reveals the moments of the limiting spectral measure. The techniques involved are the inversion and linearization coefficients of the orthogonal/skew-orthogonal polynomials, and the Christoffel-Darboux type formula. This talk is based on joint work with Gernot Akemann and Sung-Soo Byun.

Yoochan Han:  Spectral properties and weak detection in stochastic block models

Abstract.  We consider the spectral properties of balanced stochastic block models of which the average degree grows slower than the number of nodes (sparse regime) or proportional to it (dense regime). For both regimes, we prove a phase transition of the extreme eigenvalues of SBM at the Kesten-Stigum threshold. We also prove the central limit theorem for the linear spectral statistics for both regimes. We propose a hypothesis test for determining the presence of communities of the graph, based on the central limit theorem for the linear spectral statistics.

Jinyoung Kim:  Parameter inference of Chemical Reaction Networks based on high-frequency observations of species copy numbers

Abstract.  Chemical Reaction Networks provide a fundamental framework for modeling the stochastic dynamics of biochemical systems, where molecular species evolve through discrete and random noise reaction events. Parameter inference in Chemical Reaction Networks is a central problem in systems biology, but traditional methods such as maximum likelihood estimation are often intractable due to computational complexity and the lack of continuous-time data. In this study, we introduce a statistically grounded and computationally efficient estimator for reaction rate parameters using high-frequency discrete-time observations. Modeling the system as a Continuous-Time Markov Chain, our method handles general kinetics, including non-massaction and higher-order reactions. Validation on synthetic and experimental datasets demonstrates its accuracy and robustness. This approach offers a simple and reliable framework for parameter inference in complex stochastic systems.

Contributed Talks (6/24 Tue)

Yong-Woo Lee:  The number statistics for the real eigenvalues of the real Ginibre matrices

Abstract.  By nature of being real random matrices, the Ginibre orthogonal ensemble (GinOE) typically features mix of real eigenvalues and complex eigenvalue pairs, despite lack of symmetric structure. In this talk, we focus on the number of real eigenvalues of the GinOE. We present sharp asymptotic results on the probability observing exceptionally large numbers of real eigenvalues, and outline a method used to derive these results. Our approach could be generalized to the elliptic GinOE, which interpolates between the GinOE and the Gaussian orthogonal ensemble. This is based on a joint work with Gernot Akemann and Sung-Soo Byun.

Eui Yoo:  Three topological phases of the elliptic Ginibre ensembles with a point charge

Abstract.  In the large N limit, random matrix models exhibit limiting spectra in the complex plane whose support is called the droplets.

In this contributed talk, we will introduce equilibrium measure in random matrix theory as a LLN type limit and significance of the topology of the droplet in CLT type fluctuation.  As an example of a random matrix model rich in topologies of droplets, we introduce our work about  the conditional elliptic Ginibre matrix model.  We prove that the droplets of the model are either simply connected, doubly connected, or composed of two simply connected components. Moreover, we present the explicit description of the droplet and electrostatic energies for the simply and doubly connected case. This is based on a joint work with Sung-Soo Byun. (arXiv:2502.02948)

Taegyun Kim:  New nuniversality class for mixed p spin spherical model with heavy tail interaction: Free energy, limiting spin law, ultrametricity breaking

Abstract.  In this presentation, we generalize the discussion of spin glass theory to heavy tail interaction for mixed p spin spherical models. We present how ultrametricity, RSB structure and free energy fluctuation changes if we change the Gaussian interaction to heavy tail interaction in a mixed p spin spherical model. This shows novel probabilistical phase transition and breaking of ultrametricity which is core property of spin glass theory.