Hong Kong Probability Seminar

Coordinators: Zhigang Bao (HKU), Maximilian Nitzschner (HKUST), Pierre Nolin (CityU)

Organizers:

• • CityU: Yukun He, Pierre Nolin, Wei Qian

  • CUHK: Xiao Fang, Phillip Yam

  • HKU: Zhigang Bao, Guangyue Han

  • HKUST: Maximilian Nitzschner, Ke Wang, Xinghua Zheng

Past coordinator: Jianfeng Yao

Presentation

Probabilists in Hong Kong are quite dispersed in several universities, and they have had noticeable difficulty to meet regularly for academic discussions.  The Hong Kong Probability Seminar aims at filling this gap. Each month the seminar features two talks by experts in probability theory and related fields. Each talk lasts for approximately 90 minutes: such  extended lecture time, together with the informal style of the seminar and a coffee break between talks, should ease interactions among the participants.

Year 2024–2025

• • • April 10, 2025 (HKUST)

      • • 2:00–3:30pm: Colin McSwiggen (Academia Sinica)

Large deviations and multivariable special functions

Abstract: This talk introduces techniques for using the large deviations of interacting particle systems to study the large-N asymptotics of generalized Bessel functions. These functions arise from a versatile approach to special functions known as Dunkl theory, and they include as special cases most of the spherical integrals that have captured the attention of random matrix theorists for more than two decades. I will give a brief introduction to Dunkl theory and then present a result on the large-N limits of generalized Bessel functions, which unifies several results on spherical integrals in the random matrix theory literature. These limits follow from a large deviations principle for radial Dunkl processes, which are generalizations of Dyson Brownian motion. I will also discuss applications to asymptotic representation theory. All results are joint work with Jiaoyang Huang.

• • • • • 3:30–4:00pm: coffee break

        • 4:00–5:30pm: Cheuk Yin Lee (CUHK-SZ)

Strong local nondeterminism of SPDEs

Abstract: Many classical results for the Brownian motion rely on its special properties such as independent increments and Markov property. When extending beyond the Brownian motion, strong local nondeterminism (SLND) becomes a useful tool for studying more general Gaussian processes and Gaussian random fields. This applies to solutions to stochastic partial differential equations (SPDEs) driven by Gaussian noise. In this talk, I will provide an overview of SLND and present some recent results on SLND of SPDEs and their applications. I will also talk about exceptional points of SPDEs involving exceptionally large and small increments.

Venue:  Room 2405 (Lifts 17-18), Academic Building, HKUST

• • • February 10, 2025 (HKUST)

      • • 2:00–3:30pm: Jiacheng Zhang (CUHK)

Itô's formula for flows of conditional measures on semimartingales

Abstract: Motivated by recent development of mean-field systems with common noise, this paper establishes Itô’s formula for flows of conditional probability measures under a common filtration associated with general semimartingales. This generalizes existing works on flows of conditional measures on Itô processes and flows of deterministic measure on general semimartingales. The key technical components involve constructing conditional independent copies and establishing the equivalence between stochastic integrals with respect to the conditional law of semimartingales and the conditional expectation of stochastic integrals with respect to copies of semimartingales. Itô’s formula is then established for cylindrical functions through conditional independent copies, and extended to the general case through function approximations.

• • • • • 3:30–4:00pm: coffee break

        • 4:00–5:30pm: Alberto Chiarini (University of Padova)

Gradient estimates for the Schrödinger potentials and convergence to the Brenier map

Abstract: The classical Schrödinger problem is that of finding the most likely evolution of a system of independent Brownian motions conditionally on the observation of the system configuration at two deterministic times. If one assumes a very large number of particles, the resulting evolution can be recast as a large deviation question: by Sanov’s theorem the most likely evolution is found by minimizing the relative entropy with respect to the Wiener measure among all probability measures on path space with given marginal laws at the two times. The last decade has seen an upsurge in interest around the Schrödinger problem since it can be seen as a regularised version of the optimal transport problem which leads to major computational advantages. In this talk, after introducing the Schrödinger problem, we focus on the convergence of the gradients of the Schrödinger potentials to the Brenier map in the small-time limit.

Venue:  LT-B, Academic Building, HKUST

• • • December 17, 2024 (HKU)

      • • 2:00–3:30pm: Shuangping Li (Stanford)

Some easy optimization problems have the overlap-gap property

Abstract: We show that the shortest s-t path problem has the overlap-gap property in (i) sparse G(n,p) graphs and (ii) complete graphs with i.i.d. exponential edge weights. Furthermore, we demonstrate that in sparse G(n,p) graphs, shortest path is solved by O(log n)-degree polynomial estimators, and a uniform approximate shortest path can be sampled in polynomial time. This constitutes the first example in which the overlap-gap property is not predictive of algorithmic intractability for a (non-algebraic) average-case optimization problem. This is based on joint work with Tselil Schramm.

• • • • • 3:30–4:00pm: coffee break

        • 4:00–5:30pm: Luca Lionni (ENS Lyon)

Introduction to invariant random tensors

Abstract: I will discuss the characterization of invariant random tensors: their moments for finite size and asymptotically in the limit of infinite size, and known results for the real and complex, symmetric and non-symmetric Gaussians, and if time allows, the relation to the enumeration of pseudomanifolds.

Venue:  Room 210, Run Run Shaw Building, HKU

• • • October 29, 2024 (HKUST)

      • • 2:00–3:30pm: Gefei Cai (Peking University)

Solving the Three-point Connectivity for 2D Critical Percolation

Abstract: In 2D critical Bernoulli percolation, the n-point connectivity $P_n(x_1,…,x_n)$ is defined to be the probability that $x_1,…,x_n$ are all connected through open edges. It was conjectured by physicists Delfino-Viti '10 that the ratio $R=P_3(x_1,x_2,x_3)/\sqrt{P_2(x_1,x_2)P_2(x_1,x_3)P_2(x_2,x_3)}$ will converge to a universal constant, independent of $x_1,x_2,x_3$, which can be expressed through the imaginary DOZZ formula. This formula gives R ≈ 1.022 and matches the numerical simulation. In this talk we rigorously prove this conjecture, from the coupling results of Liouville quantum gravity and conformal loop ensemble developed recently. Our approach also solves 3-point connectivities of 2D critical FK-q percolations and 3-point connectivity of spin clusters in 2D critical Ising model. Based on the joint work with Morris Ang (UCSD), Xin Sun (BICMR) and Baojun Wu (BICMR).

• • • • • 3:30–4:00pm: coffee break

        • 4:00–5:30pm: Carsten Chong (HKUST)

Do Equity and Options Markets Agree about Volatility?

Abstract: In other words: Are P and Q equivalent? Here P is the physical probability measure and Q is the risk-neutral probability measure. According to the fundamental theorem of asset pricing, the answer should be yes, provided that arbitrage opportunities do not exist. However, in recent years, selling zero-date put options (i.e., put options that expire on the same day as they are traded) have been documented as an extremely profitable strategy, raising the question whether arbitrage opportunities are really absent in this new and rapidly evolving market of zero-date options. In order to address this question, we show that P-Q equivalence implies tight bounds on the Radon-Nikodym derivative of Q with respect to P and that violations of such restrictions lead to local arbitrage opportunities that can be exploited using a static portfolio of zero-date options and a dynamic position in the underlying asset. These local arbitrage opportunities are characterized by arbitrarily high reward-to-risk ratios and cause local explosion of conditional moments of the Radon-Nikodym derivative. Empirically, we find no evidence of such local arbitrage opportunities. Thus, zero-date options are expensive but not mispriced. This is based on joint work with Viktor Todorov (Northwestern).

Venue:  Room 3598 (Lifts 27-28), Academic Building, HKUST

Year 2023–2024

• • • August 9, 2024 (HKUST)

      • • 2:00–3:30pm: Daisuke Shiraishi (Kyoto University)

Loop-erased random walk in three dimensions

Abstract: Loop-erased random walk (LERW) is a random simple path obtained through erasing all loops chronologically from a simple random walk path. In this talk, I will first begin with an overview of LERW. Then I will discuss recent progress on the convergence of 3D LERW in the natural parametrization. This is joint work with Xinyi Li and Sarai Hernandez-Torres.

• • • • • 3:30–4:00pm: coffee break

        • 4:00–5:30pm: Wai-Kit Lam (NTU, Taipei)

On the geometry of the growing ball of first-passage percolation

Abstract: In first-passage percolation, one puts i.i.d. nonnegative weights on the nearest-neighbor edges of $\mathbb{Z}^d$, and studies the induced (pseudo)metric. It is well-known that a metric ball of radius $t$, after rescaled by $1/t$, converges in a certain sense to a deterministic limit shape as $t\to\infty$. However, not too much about the metric ball is known for finite $t$. In this talk, we will discuss the geometry of a ball with a large radius, and we will focus on the boundary of such a ball. We will first talk about the size of the boundary of a ball (based on an earlier joint work with M. Damron and J. Hanson), and then we will talk about holes in a ball (based on a more recent joint work with M. Damron, J. Gold and X. Shen).

Venue:  Room 6573 (Lifts 29-30), Academic Building, HKUST

• • • July 22, 2024 (CityU)

      • • 2:00–3:30pm: Dmitry Belyaev (University of Oxford)

Level sets of smooth Gaussian fields

Abstract: In this talk I will survey the theory of level sets of smooth Gaussian fields and their conjectured connection to the percolation theory. I will explain a range of conjectures and will talk about recent progress in understanding large-scale properties of level sets and sets where the field exceeds a given level. The talk is based on joint work with M. McAuley and S. Muirhead.

• • • • • 3:30–4:00pm: coffee break

        • 4:00–5:30pm: Zhigang Bao (HKUST)

Phase transition of eigenvector for spiked random matrices

Abstract: In this talk, we will first review some recent results on the eigenvectors of random matrices under fixed-rank deformation, and then we will focus on the limit distribution of the leading eigenvectors of the Gaussian Unitary Ensemble (GUE) with fixed-rank (aka spiked) external source, in the critical regime of the Baik-Ben Arous-Peche (BBP) phase transition. The distribution is given in terms of a determinantal point process with extended Airy kernel. Our result can be regarded as an eigenvector counterpart of the BBP eigenvalue phase transition. The derivation of the distribution makes use of the recently rediscovered eigenvector-eigenvalue identity, together with the determinantal point process representation of the GUE minor process with external source. This is a joint work with Dong Wang (UCAS).

Venue:  Yeung Kin Man Academic Building (Academic 1), Room B6605 (Blue zone, 6th floor), CityU. Preregistration is required for non-CityU participants: please send an email beforehand to bpmnolin@cityu.edu.hk.

• • • June 24, 2024 (HKUST)

      • • 2:00–3:30pm: Kyeongsik Nam (KAIST)

Universality of log-correlated fields

Abstract: It is widely conjectured that Poisson-Dirichlet behavior universally appears for a large class of low-temperature disordered systems. However, this principle has been verified only for particular models which are exactly solvable. In this talk, I will talk about universal behaviors of general log-correlated Gaussian fields. Based on the joint work with Shirshendu Ganguly.

• • • • • 3:30–4:00pm: coffee break

        • 4:00–5:30pm: Satomi Watanabe (CityU)

Random walk on uniform spanning tree and loop-erased random walk

Abstract: Random walks on random graphs are associated with diffusion phenomena in disordered media. This talk focuses on the simple random walk on uniform spanning trees (UST) and loop-erased random walks (LERW). The first two results are a quantitative estimate of the number of collisions and heat kernel fluctuations for the simple random walk on the three-dimensional UST. Next, I will discuss the annealed off-diagonal heat kernel of the simple random walk on high-dimensional LERWs.

Venue:  Room 4504 (Lifts 25-26), Academic Building, HKUST

• • • February 27, 2024 (HKUST)

      • • 2:30–4:00pm: Stefan Junk (Gakushuin University)

Strong disorder and very strong disorder are equivalent for directed polymers

Abstract: We consider the directed polymer model, which describes paths affected by a random space-time environment in spatial dimension d>2. The model undergoes a phase transition between a high-temperature, weak disorder phase and a low-temperature, strong disorder phase, which is characterized by whether the associated (normalized) partition function converges to zero. From the physical point of view, it is more natural to consider a very-strong-disorder regime, characterized by whether the partition function converges to zero exponentially fast. It has been a long-standing conjecture that these notions are equivalent and we now give a proof of this. Moreover, our proof reveals that weak disorder holds at the critical value. Joint work with Hubert Lacoin.

• • • • • 4:00–4:30pm: coffee break

        • 4:30–6:00pm: Yukun He (CityU)

Edge statistics of random graphs

Abstract: I will talk about some recent developments of extreme eigenvalue statistics of random graphs. The models include random regular graphs, Erdős–Rényi graphs, and Erdős–Rényi directed graphs.

Venue:  Room 4475 (Lifts 25-26), Academic Building, HKUST

• • • February 2, 2024 (CityU)

      • • 2:00–3:30pm: Mo Dick Wong (Durham University)

What can we hear about the geometry of an LQG surface?

Abstract: The Liouville quantum gravity (LQG) surface, formally defined as a 2-dimensional Riemannian manifold with conformal factor being the exponentiation of a Gaussian free field, is closely related to random planar geometry as well as scaling limits of models from statistical mechanics. In this talk, I shall explain the Weyl's law for the eigenvalues associated to the formal Laplace-Beltrami operator using Liouville Brownian motion, the canonical diffusion process on an LQG surface. This is a joint work with Nathanael Berestycki.

• • • • • 3:30–4:00pm: coffee break

        • 4:00–5:30pm: Marcin Lis (Vienna University of Technology)

On height functions and continuous spin models

Abstract: We revisit the classical phenomenon of duality between random integer-valued height functions with positive definite potentials and abelian spin models with O(2) symmetry. We use it to derive new results in quite high generality including: a universal upper bound on the variance of the height function in terms of the Green's function (a GFF bound) which among others implies localisation on transient graphs; monotonicity of said variance with respect to a natural temperature parameter; the fact that delocalisation of the height function implies a BKT phase transition in planar models; and also delocalisation itself for height functions on periodic "almost" planar graphs. This is joint work with Diederik van Engelenburg.

Venue:  Yeung Kin Man Academic Building (Academic 1), Room B6605 (Blue zone, 6th floor), CityU. Preregistration is required for non-CityU participants: please send an email beforehand to bpmnolin@cityu.edu.hk.

• • • November 23, 2023 (HKUST)

      • • 2:00–3:30pm: Yuanyuan Xu (AMSS, CAS)

Universality of extreme eigenvalues of a large non-Hermitian random matrix

Abstract: We will report on recent progress regarding the universality of the extreme eigenvalues of a large random matrix with i.i.d. entries. Beyond the radius of the celebrated circular law, we will establish a precise three-term asymptotic expansion for the largest eigenvalue (in modulus) with an optimal error term. Based on this result, we will further show that the properly normalized largest eigenvalue converges to a Gumbel distribution as the dimension goes to infinity. Similar results also apply to the rightmost eigenvalue of the matrix. These results are based on several joint works with Giorgio Cipolloni, László Erdős, and Dominik Schröder.

• • • • • 3:30–4:00pm: coffee break

        • 4:00–5:30pm: Maximilian Nitzschner (HKUST)

Bulk deviation lower bounds for the simple random walk

Abstract: In this talk we present large deviation lower bounds for the probability of certain bulk- deviation events depending on the occupation-time field of a simple random walk on the Euclidean lattice in dimensions larger or equal to three. As a particular application, these bounds imply an exact leading order decay rate for the probability of the event that a simple random walk covers a substantial fraction of a macroscopic body, when combined with a corresponding upper bound previously obtained by Sznitman. As a pivotal tool for deriving such optimal lower bounds, we recall the model of tilted walks which was first introduced by Li in order to develop similar large deviation lower bounds for the probability of disconnecting a macroscopic body from an enclosing box by the trace of a simple random walk. We then discuss a refined local coupling with the model of random interlacements which is used to locally approximate the occupation times of the tilted walk. Based on joint work in progress with A. Chiarini (University of Padova).

Venue:  Room 3598 (Lifts 27-28), Academic Building, HKUST

Year 2022–2023

• • • May 18, 2023 (CityU)

      • • 2:00–3:30pm: Manan Bhatia (MIT)

Duality and fractality in the directed landscape

Abstract: We survey recent developments around the fractal structure of the directed landscape – a canonical continuum model of two dimensional random geometry known to be scaling limit of exponential last passage percolation. This model, like other models of random geometry, exhibits the phenomenon of geodesic confluence, wherein geodesics started from a pair of nearby points and going to a distant pair of nearby points quickly merge with each other, travel along a highway, and then split to go to their respective destinations. We discuss some general useful techniques including difference profiles, the connection of disjoint optimizers and the Airy line ensemble, and the Brownian Bessel decomposition around geodesics, and indicate how these have be used to study various questions around the fractal structure of atypical stars—a measure zero set of points where geodesic confluence fails. We also discuss the duality between semi-infinite geodesics and competition interfaces present in discrete last passage percolation and its consequences regarding the fractal structure of the continuum tree formed by semi-infinite geodesics. Finally, we look at the space-filling Peano curve between the dual pair of geodesic trees in the continuum, recently constructed in a joint work with Basu, and discuss its regularity and fractal properties.

• • • • • 3:30–4:00pm: coffee break

        • 4:00–5:30pm: Wei Qian (CityU)

Geodesics in the Brownian map and the Brownian disk

Abstract: This talk surveys properties of geodesics in various Brownian surfaces with or without a boundary. We first give some general background, and then present a strong and quantitative form of the confluence of geodesics phenomenon which states that any pair of geodesics which are sufficiently close in the Hausdorff distance must coincide with each other except near their endpoints.

We then deduce various existence and dimension results about geodesics in the Brownian map and Brownian disk. In particular, we classify the (finite number of) possible configurations of geodesics between any pair of points, up to homeomorphism, and give a dimension upper bound for the set of endpoints in each case.

Finally, we show that every geodesic can be approximated arbitrarily well and in a strong sense by a geodesic connecting typical points. In particular, this gives an affirmative answer to a conjecture of Angel, Kolesnik, and Miermont that the geodesic frame, the union of all of the geodesics in the Brownian map minus their endpoints, has dimension one, the dimension of a single geodesic.

This talk is based on a joint work with Jason Miller, and joint works in progress with Manan Bhatia and with Tiancheng He.

Venue:  Yeung Kin Man Academic Building (Academic 1), Room B6605 (Blue zone, 6th floor), CityU. Preregistration is required for non-CityU participants: please send an email beforehand to weiqian@cityu.edu.hk.

• • • February 24, 2023 (HKUST)

      • • 2:00–3:30pm: Antoine Jego (EPFL, Switzerland)

Conformally invariant fields out of Brownian loop soups

Abstract: The Brownian loop soup in some two-dimensional domain is a random collection of infinitely many Brownian-type loops. It is sampled according to theta times a certain loop measure, where theta > 0 is a parameter of the model. First introduced by Lawler and Werner, it has become a central object of study in random conformal geometry, in particular due to its connections with conformal loop ensembles (CLE) and the Gaussian free field (GFF). This latter connection, referred to as Le Jan’s isomorphism, states that, when theta = 1/2, the occupation field of the loop soup has the same law as the square of a GFF. Remarkably, it is even possible to fully recover the GFF (also its sign) out of the Brownian loop soup with the help of additional coin tosses (Lupu and Aru—Lupu—Sepulveda).

The main goal of this talk is to answer the following question: What is the field naturally associated to the loop soup when theta < 1/2? Along the way, we will have to study the percolative properties of the loop soup. 

• • • • • 3:30–4:00pm: coffee break

        • 4:00–5:30pm: Yifan Gao (CityU)

Multiple points on the boundaries of Brownian loop-soup clusters

Abstract: The Brownian loop soup, introduced by Lawler and Werner in 2004, plays a crucial role in understanding two-dimensional models in statistical physics. In this talk, I will present the fractal property of the Brownian loop soup by the Hausdorff dimension of multiple points on the boundaries of clusters. The first part of the talk is to review the well-known results on the Hausdorff dimension of special points of the Brownian motion, which is intimately related to a family of exponents. In the second part, I will explain how to obtain our result via generalizing such exponents to the Brownian loop soup. The challenge is to prove a corresponding separation lemma which is unclear due to the interplay of Brownian motions and the Brownian loop soup. I will highlight this powerful lemma with a final remark on its use in our future work. Based on a joint work (arXiv:2205.11468) with Xinyi Li (Peking University) and Wei Qian (City University of Hong Kong).

Venue:  Room 1511 (Lifts 27-28), Academic Building, HKUST

Year 2018–2019

• • • April 26, 2019 (HKU)

      • • 2:00–3:30pm: Piotr Graczyk (Université d'Angers)

Empirical processes of particle systems

• • • • • 3:30–4:00pm: coffee break

        • 4:00–5:30pm: Wei Qian (University of Cambridge)

Decomposition of Brownian loop-soup clusters in dimension two

Abstract: Brownian loop-soups were introduced by Lawler and Werner in 2004 as a Poisson point process of Brownian loops. In dimension two, their distributions are invariant under conformal maps. They are also intimately related to other important random objects that I will first present, such as the Gaussian free field and the conformal loop ensembles. We will then focus on the decomposition of Brownian loop-soup clusters (connected components of loops). In particular, we obtain a surprising decomposition for clusters at the critical intensity in terms of Poisson point processes of Brownian excursions. A large part of this talk is based on joint works with W. Werner.

Venue:  Room 210, Run Run Shaw Building, HKU      (The event is supported by the Institute of Mathematical Research, Department of Mathematics, The University of Hong Kong).

• • • March 15, 2019 (HKUST)

      • • 2:00–3:30pm: Michael Choi (CUHK Shenzhen)

Accelerated simulated annealing under fast cooling

Abstract: Originated from statistical physics, simulated annealing is a popular stochastic optimization algorithm that has found extensive empirical success in disciplines ranging from image processing to statistics and combinatorial optimization. At the heart of the algorithm lies in constructing a non-homogeneous Markov process that converges to the set of global minima as the temperature cools down. In this talk, we will first review the classical theory for simulated annealing and discuss some of its theoretical limitations. We will then introduce a promising accelerated variant of simulated annealing that provably converges faster and does not suffer from the drawbacks of its classical counterpart. This talk is based on http://arxiv.org/abs/1901.10269.

• • • • • 3:30–4:00pm: coffee break

        • 4:00–5:30pm: Leonardo T. Rolla (U. Buenos Aires and NYU Shanghai)

Absorbing-state phase transitions

Abstract: Modern statistical mechanics offers a large class of driven-dissipative stochastic systems that naturally evolve to a critical state, of which Activated Random Walks is one of the best examples. The main pursuit in this field is to describe the critical behavior, scaling relations and critical exponents of such systems, and the connection between driven-dissipative dynamics and conservative dynamics in infinite space. The study of this model has challenged mathematicians for a long time. We will present the partial progress made during the last ten years. These covered most of the questions regarding existence of an absorbing and an active phase for different ranges of parameters, and current efforts are drifting towards the description of critical states, scaling limits, etc. We will summarize the current state of art and discuss some of the many open problems.

Venue:  Room 4475 (via Lifts 25/26)​, Academic Building, HKUST      (The event is supported by the  Department of Mathematics,  Hong Kong University of Science and Technology).

• • • February 14, 2019 (CUHK)     (Note the unusual day: Thursday!)

      • • 2:00–3:30pm: Qi-Man Shao (CUHK)      

Wandering around the Asymptotic Theory

Abstract: In this talk we will be wandering around the asymptotic theory in probability and statistics, from the classical limit theory to recent developments and newly developed tools and techniques.

• • • • • 3:30–4:00pm: coffee break

        • 4:00–5:30pm: Federico Camia (NYU Abu Dhabi)

Limit laws and Conformal Ensembles in the planar Ising model

Abstract: In the last twenty years there has been tremendous progress in the mathematical understanding of phase transitions for models of statistical mechanics defined on planar lattices. Much of that progress is related to the study of scaling limits, obtained by sending the lattice spacing to zero. In this talk I will give a brief introduction to scaling limits and present some recent results in the mathematical theory of phase transitions. I will focus on the case of the Ising model, which was introduced in the 1920s to study ferromagnetism and is one of the most studied models of statistical mechanics. I will discuss the convergence of the Ising magnetization to a random field (i.e., a random generalized function) with interesting properties of conformal covariance, and the connection with Euclidean field theory and the associated quantum field theory. (Based on collaborations with Rene Conijn, Christophe Garban, Jianping Jiang, Demeter Kiss, and Chuck Newman.) 

Venue:  LT2, Yasumoto International Academic Park (YIA), CUHK      (The event is supported by the Department of Statistics, The Chinese University of Hong Kong). 

• • • January 11, 2019 (HKU)

• • • • • 2:00–3:30pm: Ka Chun Cheung (HKU)

On asymptotic additivity of tail risk measures

Abstract:  As perceived from daily experience together with numerous empirical studies, upper tail comonotonicity adequately describes the extremal dependence structure of multivariate risks especially over the course of financial turmoils or industrial accidents and outbreaks. Under this dependence structure, we establish the universal asymptotic additivity, as the probability level approaching to 1, for both Value-at-Risk and Conditional Tail Expectation for a portfolio of risks, in which each marginal risk could be any one having a finite endpoint or belonging to one of the three maximum domains of attraction. This covers most distributions commonly encountered in practice. Our results do not require the tail equivalence assumption as needed in the existing literature, and resolve a lasting problem in quantitative risk management. If time permits, results on asymptotic sub/super-additivity of tail risk measures under general Archimedean copula with regular varying generator will also be discussed.

This talk is based on a joint work with Hok Kan Ling, Qihe Tang, Phillip Yam and Fei Lung Yuen.

• • • • • 3:30–4:00pm: coffee break

        • 4:00–5:30pm: Zuoquan Xu (Poly U)

On probability distortion and applications in behavioral finance

Abstract:  In this talk, I will first introduce probability distortion/weighting function and its roles in behavioral finance theories.  Then I will describe the recent development of the so-called quantile optimization method, a main tool to deal with optimization problems involving probability weighting from financial economics. In particular, portfolio selection, optimal stopping, and insurance models will be solved by this method.

Venue:  Room 301, Run Run Shaw Building, HKU   (The event is supported by the  Department of Statistics and Actuarial Science, The University of Hong Kong).

• • • November 23, 2018 (HKUST)

      • • 1:30–2:30pm: Rongfeng Sun (National University of Singapore)

Moment asymptotics for the (2+1)-dimensional directed polymer in the critical window

Abstract: The partition function of the directed polymer model on Z^{2+1} has been shown to undergo a phase transition on an intermediate disorder scale. In this talk, we focus on a window around the critical point. Exploiting a renewal process representation, we identify the asymptotics for the second and third moments of the partition function. As a corollary, we show that, viewed as a random field, the family of partition functions admits non-trivial diffusive scaling limits, and each limit point has the same covariance structure with logarithmic divergence near the diagonal. Similar results are obtained for the stochastic heat equation on R^2, extending earlier results by Bertini and Cancrini (98). Based on joint work with F. Caravenna and N. Zygouras.

Venue: Room 2502, HKUST

• • • • • 2:30–3:00pm: coffee break (Room 2502, HKUST)

        • 3:00–4:00pm: Tomohiro Sasamoto (Tokyo Institute of Technology)

The Kardar-Parisi-Zhang (KPZ) models and their universality

Abstract: The Kardar-Parisi-Zhang (KPZ) equation is a nonlinear stochastic partial differential equation which was introduced in 1986 to describe the motion of interface. Fluctuations of the interface exhibit universal scaling laws, now known as the KPZ universality. In 2010 the exact formula for the one-point height distribution was discovered by Sasamoto-Spohn and Amir-Corwin-Quastel and there have been many developments since then.

In this talk, we start from explaining the basics about the KPZ equation and its universality. We first present the equation and discuss the issue of its well-definedness. Then we show and explain how to derive the exact formula for the height distribution, and study its limiting behaviors.

Then we discuss various recent developments on the topic. They include the introduction and analysis of various lattice models in the KPZ universality, the connections to integrable systems and representation theory, and generalizations to multi-component systems. Finally we also mention a few outstanding problems on the subject.

Venue: Lecture Theater F, HKUST      (This talk is joint with the departmental colloquium)

• • • November 2, 2018 (CUHK)

      • • 2:00–3:30pm: Jie Xiong (Southern University of Science and Technology)

Stackelberg game with partial information

Abstract: Motivated by the cooperative advertising and pricing problems, we consider the leader-follower game with asymmetric information. As preparation, I will first introduce the theory of nonlinear filtering which is one of the main tool used in this research. After that we consider the general stochastic maximum principles under partial information when the state is given by a BSDE or an FBSDE with or without mean-field term. After these preparation, we will discuss the stochastic game under asymmetric information structure.

• • • • • 3:30–4:00pm: coffee break

        • 4:00–5:30pm: Chen Wang (HKU)

Some new results on random matrix theory with application to analysis of dynamic factor models

Abstract: This talk consists of two parts. In the first part, I will give a brief introduction to some relevant results of random matrix theory (RMT). The second part will focus on a specific application, that is, the order determination of large dimensional dynamic factor model.

Venue:  LT1, Lady Shaw Building, CUHK      (The event is supported by the Department of Statistics, The Chinese University of Hong Kong).

• • • October 5, 2018 (HKU)

      • • 2:00–3:30pm: Phillip Yam (CUHK)

Beyond classical portfolio selection

Abstract: Since the first introduction in Markowitz (1952), portfolio choice theory has been one of the key research topics in mathematical finance, and it is a formal one on striving for an ideal balance between the portfolio return and reducing its inherent risk inherited from various financial markets and operations. Yet, with the increasing sophistication of different markets, even in the presence of notable behavioral bias of investors, there is an urgent call for reframing the landscape of this traditional research area in response to new desire. Based on some of my recent research effort, I shall aim to share with my view on some possible new directions that can cater those practical considerations.

• • • • • 3:30–4:00pm: coffee break

        • 4:00–5:30pm: Pierre Tarrès (NYU Shanghai)

Self-interacting random walks and statistical physics

Abstract: We start by a review of recent questions and results on self-interacting random walks. Then we explain how the Edge-reinforced random walk, introduced by Coppersmith and Diaconis in 1986, is related to several models in statistical physics, namely the supersymmetric hyperbolic sigma model studied by Disertori, Spencer and Zirnbauer (2010), the random Schrödinger operator and Dynkin's isomorphism.

These correspondences enable us to show recurrence/transience results on the Edge-reinforced random walk, and they also allow us to provide insight into these models. This work is joint with Christophe Sabot, and part of it is also in collaboration with Margherita Disertori, Titus Lupu and Xiaolin Zeng.

Venue:  Room 103, Run Run Shaw Building, HKU      (The event is supported by the Institute of Mathematical Research, Department of Mathematics, HKU).

Year 2017–2018

• • • • April 27, 2018 (CUHK)

        • • 2:00–3:30pm: Ke Wang (HKUST)

Random perturbation of low-rank matrices and applications

Abstract: Computing the singular values and singular vectors of a large matrix is a basic task in high dimensional data analysis with many applications in computer science and statistics. In practice, however, data is often perturbed by noise. It is naturable to understand the essential spectral parameters of this perturbed matrix, such as its spectral norm, the leading singular values, and vectors, or the subspace formed by the first few singular vectors. Classical (deterministic) theorems, such as those by Davis-Kahan, Wedin, and Weyl, give tight estimates for the worst-case scenario. In this talk, I will consider the case when the perturbation is random. In this setting, better estimates can be achieved when the data matrix has low rank. I will also discuss some applications of our results. This talk is based on joint works with Sean O'Rourke and Van Vu.

• • • • • • 3:30–4:00pm: coffee break

          • 4:00–5:30pm: Lihu Xu (University of Macau)

Approximation of stable law in Wasserstein distance by Stein's method

Abstract: We will first give a fast review of some preliminaries of stable law, stable processes, ergodicity of SDEs driven by stable noises, and then talk how to obtain the convergence rate of stable law in Wasserstein distance by Stein's method. If the time is permitted, we will give a sketch on using a method recently developed by Fang, Shao and Xu to sample high dimensional stable distribution by discretizing Ornstein-Uhlenbeck stable processes. This talk is based on the paper arXiv:1709.00805 and a joint work in progress with Peng Chen (PhD student at UM) and Ivan Nourdin (Luxembourg).

Venue:  LT3, Lady Shaw Building, CUHK (The event is supported by the Department of Statistics, The Chinese University of Hong Kong).

• • • March 23, 2018 (HKUST)

      • • 2:00–3:30pm: Benoît Collins (Kyoto University)

Strong convergence for random permutations

Abstract: We consider an n dimensional random matrix model obtained from a non-commutative polynomial in d unitaries and their inverse, after replacing the formal unitaries by random iid permutation matrices. This model has an obvious (Perron Frobenius) eigenvector and leaves invariant its orthogonal. We study the large n limit behavior of this model on the orthogonal of the PF eigenvector and show that in addition to asymptotic freeness, it has asymptotically no outliers. Time allowing, we will also discuss applications to random graph theory.

This is joint work with Charles Bordenave.

• • • • • 3:30–4:00pm: coffee break

        • 4:00–5:30pm: Laurent Ménard (Université Paris Ouest Nanterre)

Random planar triangulations with an Ising model

Abstract: Angel and Schramm proved in 2003, that uniform planar triangulations converge for the local topology. The limit law, known as UIPT (for Uniform Infinite Planar Triangulation) has been much studied since and is now a well understood object.

In this talk, I'll explain how such objects are defined and studied. In particular, I'll explain why the algebraicity of the generating functions is crucial, and where it comes from.

I'll then turn to triangulations weighted by an Ising model and show how to extend the combinatorial results known for uniform triangulations and the local weak convergence.

This is a joint work with Marie Albenque and Gilles Schaeffer.  

Venue:  Room 4475 (via Lifts 25/26)​, Academic Building, HKUST.

• • • March 2, 2018 (HKU)

      • • 2:00–3:30pm: Tze Leung Lai (Stanford)

MCMC with sequential state substitutions: theory and applications

Abstract: Motivated by applications to adaptive filtering that involves joint parameter and state estimation in hidden Markov models, we describe a new approach to MCMC, which uses sequential state substitutions for its Metropolis-Hastings-type transitions. The basic idea is to approximate the target distribution by the empirical distribution of N representative atoms, chosen sequentially by an MCMC scheme so that the empirical distribution converges weakly to the target distribution as the number K of iterations approaches infinity. Making use of coupling arguments and bounds on the total variation norm of the difference between the target distribution and the empirical measure defined by the sample paths of the MCMC scheme, we develop its asymptotic theory. In particular, we establish the asymptotic normality (as both K and N become infinite) of the estimates of functionals of the target distribution using the new MCMC method, provide consistent estimates of their standard errors, and derive oracle properties that prove their asymptotic optimality. Implementation details and applications, particularly to adaptive particle filtering with consistent standard error estimate, are also given.

• • • • • 3:30–4:00pm: coffee break

        • 4:00–5:30pm: Chung Yin Amy Pang (HKBU)

Lumpings of algebraic Markov chains arise from subquotients

Abstract: A function on the state space of a Markov chain is a “lumping” if observing only the function values gives a Markov chain. I will describe some classical examples of lumping, for some card-shuffling models, then explain how these lumpings can be proved in a uniform way through the framework of “algebraic Markov chains”. This talk is based on Part I of the preprint of the same title.  

Venue:  Room 210, Run Run Shaw Building, HKU   (The event is supported by the Institute of Mathematical Research, Department of Mathematics, The University of Hong Kong).

• • • January 26, 2018 (CityU)

      • • 2:00–3:30pm: Xiao Fang (CUHK)

An introduction to Stein’s method

Abstract: Stein's method is a powerful tool for proving limit theorems along with error bounds. The method works for both normal and non-normal approximations for random variables with various dependency structures. In this talk, I will give a brief introduction to Stein's method. 

• • • • • 3:30–4:00pm: coffee break

        • 4:00–5:30pm: Jianfeng Yao (HKU)

On central limit theorems for eigenvalue statistics of a large Wigner matrix 

Abstract: In this talk, I will first review some well established central limit theorems for eigenvalues of a large Wigner matrix. The focus will then be on a CLT given in Bai and Yao (Bernoulli, 2005) with a detailed description of the main tools and steps of its proof. In the second part of the talk, I will discuss a related problem we recently studied for the adjacency matrix of a large random graph from the so-called “stochastic block model”.

Venue:  Yeung Kin Man Acad Building (Academic 1), Room Y5-305 (Yellow zone, 5th floor), CityU 

• • • December 15, 2017 (HKUST)

      • • 2:00–3:30pm: Guangyue Han (HKU)

An introduction to information theory

Abstract: In this talk, I will give a brief introduction to information theory and talk about some open problems and certain research directions.

• • • • • 3:30–4:00pm: coffee break

        • 4:00–5:30pm: Xinghua Zheng (HKUST)

Spatial SIR epidemic processes and their asymptotics

Abstract: We focus on a special interacting particle system known as SIR epidemic model. We will discuss its connection with branching random walk (and percolation), its measure-valued limiting process at or near criticality, and the associated phase transition phenomena. The talk is  based on joint works with Steve Lalley, Ed Perkins and Eyal Neuman.

Venue:  Lecture Theatre K (LTK), Academic Building, HKUST

• • • November 17, 2017 (CUHK)

      • • 2:00–3:00pm: István Berkes (TU Graz)

Fluctuations of stochastic processes and strong invariance principles

Abstract: Describing the fluctuations of stochastic processes over short intervals is a basic problem of probability theory with numerous applications in statistics. For example, to detect short term, "epidemic" changes in the structure of time series requires studying the fluctuations of the partial sum process of the sample (X1, X2, ... , Xn) over intervals of length very short compared with n. If X1, X2, ... are i.i.d. Gaussian variables, such results are available from the theory of Wiener process and using the celebrated Komlós-Major-Tusnády (KMT) approximation theorem, these results can be extended to the general i.i.d. case. For the case of weakly dependent sequences, a class covering many important applications, the KMT theorem is not available, except for a few special cases settled recently (Berkes-Liu-Wu 2014, Merlevède and Rio 2015). The purpose of this talk to show that using a simple modification of the elementary Bernstein blocking technique combined with the original KMT result, we can get widely applicable fluctuation results for many weakly dependent models, such as mixing processes, Markov processes, Gaussian processes, etc.

• • • • • 3:00–3:30pm: coffee break

        • 3:30–5:00pm: Jian Song (HKU)

Long-term asymptotics for (fractional) Anderson models

Abstract: In this talk, I will review our recent results on the long-term behavior of the solutions to the parabolic and hyperbolic Anderson models. The talk will consist of two parts. The first part concerns the existence and uniqueness of the solutions to the (fractional) heat equation and wave equation driven by multiplicative Gaussian noise, and the Feynmnan-Kac formula for stochastic heat equation. The second part deals with moments Lyapunov exponents for the solutions to the (fractional) Anderson models.

Venue:  LT9, Yasumoto international Academic Park (YIA), CUHK (The event is supported by the Department of Statistics, The Chinese University of Hong Kong).

• • • October 20, 2017 (HKU)

      • • 2:00–3:30pm: Pierre Nolin (CityU)

Frozen percolation and self-organized criticality

Abstract: We first give a short introduction to Bernoulli percolation, which is obtained by deleting at random, independently, the edges (or the vertices) of a given lattice. It is arguably one of the simplest models from statistical mechanics displaying a phase transition, i.e. a drastic change of macroscopic behavior, at a certain critical threshold. We present the main tools and techniques used to study percolation, as well as the most important results. We then discuss the frozen percolation model, where connected components stop growing ("freeze") as soon as they become large (i.e. reach a "size" at least N, for some finite parameter N). In particular, we explain why the "near-critical" regime of Bernoulli percolation arises. This talk is based on joint works with Rob van den Berg (CWI and VU, Amsterdam) and Demeter Kiss.

• • • • • 3:30–4:00pm: coffee break

        • 4:00–5:30pm: Zhigang Bao (HKUST)

Supersymmetry method and delocalization of random block band matrices

Abstract: For large dimensional random band matrices, a famous open question is Anderson’s localization-delocalization transition for the eigenvectors, which states that the eigenvectors of the random band matrix are extended (delocalized) if the band width is larger than the square root of the matrix size, and are otherwise localized. So far, the most hopeful method to attack this question is the supersymmetry method, which is ubiquitous in physics literature. However, the rigorous justification of supersymmetry in mathematics is still notoriously difficult. In this talk, I will introduce a recent result on delocalization of random block band matrices via a rigorous supersymmetry approach. This is a joint work with László Erdös.

Venue:  Room 210, Run Run Shaw Building, HKU   (The event is supported by the Institute of Mathematical Research, Department of Mathematics, The University of Hong Kong).