Self-avoiding walk (SAW) is a walk that each path does not visit the same point more than once. It is a model of fundamental interest in combinatorics, probability theory, statistical physics and polymer chemistry. It is certainly non-Markovian. These features makes the subject difficult, and many of the central problems remain unsolved.
This series of lectures will be divided into three talks. The first talk, we first give an overview the SAW and review the lace expansion.

The lace expansion is one of the powerful tools for investigating critical phenomena. It has succeeded in obtaining asymptotic expansions for the critical point in several models, e.g., the self-avoiding walk, ordinary/oriented percolation, the contact process, etc. One of our aims is to obtain such an asymptotic expansion for the quantum Ising model, which is one of the models of ferromagnetism. In this model, Spin configurations are randomly realized by the Hamiltonian operator (energy), which involves the z-axis and x-axis Pauli matrices. In other words, the quantum Ising model is defined by applying a transverse field to the classical Ising model. Due to the transverse field, the quantum Ising model exhibits a different type of phase transition from the classical case.

In this talk, I will show the derivation of a lace expansion for the quantum Ising model and upper bounds on the expansion coefficients. The derivation is based on the random current representation [Björnberg and Grimmett (2009) J. Stat. Phys.] [Crawford and Ioffe (2010) Commun. Math. Phys.] on space-time. This representation provides the connectivity between two vertices in space-time, which is similar to percolation models. It helps us to derive the lace expansion.

This talk is based on joint work with Akira Sakai (Hokkaido University, Japan).

We consider an optimal consumption and investment problem with delay under a linear Gaussian stochastic factor model. A linear Gaussian stochastic factor model is a stochastic factor model in which the mean returns of risky assets depend linearly on underlying economic factors that are formulated as the solutions of linear stochastic differential equations. We consider the performance-related capital inflow/outflow, which implies that the wealth process is modeled by a stochastic differential delay equation. We also treat the partial information case where the investor cannot observe the factor process and can use only past information about risky assets. Under this setting, the investor tries to maximize the finite horizon discounted expected HARA utility of consumption, the terminal wealth, and the average wealth. A pair of forward-backward stochastic differential equations derived via the stochastic maximum principle have an explicit solution that can be obtained by solving a time-inhomogeneous Riccati differential equation. Thus, the optimal strategy and the optimal value can be obtained explicitly.

A log-optimal portfolio is any portfolio that maximizes the expected logarithmic growth (ELG) of an investor’s wealth, which typically assumes prior knowledge of the true return distribution. However, in practice, return distributions are often ambiguous; i.e., the true distribution is unknown, making this problem challenging to solve. This paper proposes a supporting hyperplane approximation approach, reformulating a class of distributional robust logoptimal portfolio problems with polyhedron ambiguity sets into tractable robust linear programs. An efficient algorithm is presented to determine the optimal number of hyperplanes. Additionally, to adapt to the constantly changing market, we propose an online trading algorithm using a sliding window approach to solve a sequence of robust linear programs, offering significant computational advantages. The effectiveness of the proposed approach is supported by empirical studies using historical stock price data.

We study Merton's expected utility maximization problem in an incomplete market, characterized by a factor process in addition to the stock price process, where all the model primitives are unknown. The agent under consideration is a price taker who has access only to the stock and factor value processes and the instantaneous volatility. We propose an auxiliary problem in which the agent can invoke policy randomization according to a specific class of Gaussian distributions, and prove that the mean of its optimal Gaussian policy solves the original Merton problem. With randomized policies, we are in the realm of continuous-time reinforcement learning (RL) recently developed in Wang et al. (2020) and Jia and Zhou (2022a,b, 2023), enabling us to solve the auxiliary problem in a data-driven way without having to estimate the model primitives. Specifically, we establish a policy improvement theorem based on which we design both online and offline actor–critic RL algorithms for learning Merton’s strategies. A key insight from this study is that RL in general and policy randomization in particular are useful beyond the purpose for exploration – they can be employed as a technical tool to solve a problem that cannot be otherwise solved by mere deterministic policies. At last, we carry out both simulation and empirical studies in a stochastic volatility environment to demonstrate the decisive outperformance of the devised RL algorithms in comparison to the conventional model-based, plug-in method. Joint work with Min Dai, Yuchao Dong and Yanwei Jia.

(joint work with Shuenn-Jyi Sheu and Thi-Hien Nguyen)
In this talk, I want to present some results and open questions related to the analysis of the expansion of the spectral gap
$$ \rho(c)=-\sup\{ {\rm Re}(z);z\in{\rm Spec}(L_c)\backslash\{0\}\} $$
of the family of operators
$$ L_cf=\Delta f+(-D+cC) x \nabla f ; c\in\mathbb{R} $$
as $ c\rightarrow\infty $, where $ D $ is a symmetric positive definite $ m\times m $-matrice and $ C $ is an $ m\times m $-matrice satisfying $ \langle Cx,Dx\rangle=0 $. We will see that, due to some result from Matafune, Pallara and Priola, the question can be reduced to the study of the asymptotic behavior of the eigenvalues of the matrice $ E_c=-D+cC $ as $ c\rightarrow\infty $. The special relationship between $ D $ and $ C $ enables us to obtain some iterative method for determining the coefficients of the power series expansion of the eigen values and eigen vectors of the matrix family $ E_\epsilon=-\epsilon D+C $ as $ \epsilon\rightarrow0 $. This then yields information on the speed at which the gap $ \rho_c $ converges as $ c\rightarrow\infty $. The method works well as long as the matrice $ C $ does not have eigen vectors with eigen space of dimension larger than one. Open questions remain in the general situation.

The two-dimensional stochastic heat equation (SHE) at criticality was introduced in the '90s. It arises from problems of statistical physics through several models of stochastic surface growth dynamics and the disordered system of a directed polymer in a random medium. Nevertheless, the equation has been known to pose difficulties to solution theories of stochastic partial differential equations despite the "trivial" form that the equation takes.
This talk will provide an overview of stochastic analytic descriptions of the two-dimensional SHE at criticality. Such descriptions for the discussion include the Feynman-Kac-type formulas for the moments and extend to the pathwise level by the martingale characterization of the solutions.

This talk offers a mathematical introduction to diffusion models in generative AI, based on their original formulations as presented in references [1] and [2], without introducing new insights. We will begin by discussing diffusion processes. Within the context of diffusion models, we will illustrate both the forward and reverse diffusion processes, highlighting their roles within the generative AI framework. The emphasis will be on the mathematical structure rather than intuitive explanations or theoretical justifications.

References
[1] Ho, J., Jain, A., and Abbeel, P. (2020). Denoising Diffusion Probabilistic Models.
[2] Sohl-Dickstein, J., Weiss, E. A., Maheswaranathan, N., and Surya Ganguli. (2015). Deep Unsupervised Learning using Nonequilibrium Thermodynamics. In International Conference on Machine Learning, 2256–2265

One of the main difficulties in solving stochastic partial differential equations (SPDEs) is the poor regularity of their solutions, which prevents the direct application of classical PDE techniques. However, rough path theory offers a way to solve this issue under suitable conditions. In this talk, I will introduce what rough path integration is and explain how it solves SPDEs in Martin Hairer's work. I will then introduce how this technique can be combined with the Littlewood-Paley theory to tackle our specific problem.

The primary goal of this lecture series is to introduce emerging topics in high- dimensional statistical inference. Specifically, we will focus on statistical physics models, such as the Ising model and its disordered variants. Parameterized by the temperature and external field, these models are spin systems, where the spin interactions are described by a structure matrix. While their Gibbs measures belong to the exponential family, understanding their corresponding parameter and structure estimation problems, based on a limited number of given samples, presents intriguing mathematical challenges. Throughout the series, we will highlight recent advancements, explain the approaches, and propose open problems that are particularly well-suited for graduate students and junior researchers to expand their research interests.

This talk offers a mathematical introduction to diffusion models in generative AI, based on their original formulations as presented in references [1] and [2], without introducing new insights. We will begin by discussing diffusion processes. Within the context of diffusion models, we will illustrate both the forward and reverse diffusion processes, highlighting their roles within the generative AI framework. The emphasis will be on the mathematical structure rather than intuitive explanations or theoretical justifications.

References
[1] Ho, J., Jain, A., and Abbeel, P. (2020). Denoising Diffusion Probabilistic Models.
[2] Sohl-Dickstein, J., Weiss, E. A., Maheswaranathan, N., and Surya Ganguli. (2015). Deep Unsupervised Learning using Nonequilibrium Thermodynamics. In International Conference on Machine Learning, 2256–2265

The Critical 2d Stochastic Heat Flow arises as a non-trivial solution of the Stochastic Heat Equation (SHE) at the critical dimension 2 and at a phase transition point. It is a log-correlated field which is neither Gaussian nor a Gaussian Multiplicative Chaos. We will review the phase transition of the 2d SHE, describe the main points of the construction of the Critical 2d SHF and outline some of its features and related questions. Based on joint works with Francesco Caravenna and Rongfeng Sun.

This is a mini course that aims at an audience of advanced undergraduate and graduate students as well as to researchers in probability, analysis and beyond. We will start with the robust method of the Lindeberg principle, which was originally used to prove the standard central limit theorem. We will then lift the Lindeberg principle to a setting of more complicated random objects such as multilinear polynomials and Wiener chaoses. The latter are intimately connected with recent developments in Stochastic PDEs and disordered systems and we will demonstrate how the Lindeberg principle can be used to obtain scaling limits of such models. We will then to move into nonlinear settings where chaos expansions are replaced by tree structures via the so called Butcher series. Relations to singular SPDEs will also be discussed.

Branching random walks are fundamental models in probability theory that describe the evolution of populations in which individuals move randomly and reproduce according to stochastic rules. These models arise in diverse fields, including statistical physics, evolutionary biology, and computer science. In this talk, we introduce the framework for constructing a branching random walk from a Galton-Watson branching process and explore the limiting behavior of the distribution of individuals' positions.

Branching processes are stochastic processes that are often used to model the evolution of populations. They are also widely used in probability theory, biology (e.g., population genetics, epidemiology), and other fields. In this talk, we will introduce the most fundamental type of branching process, the Galton-Watson branching process, and review some classical limit theorems that describe the long-term behavior of populations. In addition, we will also discuss the coalescence problem, which provides us with a different perspective on investigating the history of the population.

Random walks on particle systems, such as the simple exclusion process (SEP, in which particles perform simple random walks that cannot occupy simultaneously the same position), have attracted notable attention in the last decade. In such models, say on the d-dimensional lattice, a walker moves at each unit of time to a random neighbouring vertex, with a drift that depends on whether its current location is occupied or not by a particle of the environment.
Compared to standard static environments, many classical techniques break down. A recurring question is then whether features of the static world are still present in the dynamic case, such as strong trapping effects that can force a transient walk to have zero speed. This is in particular relevant for the SEP, which is conservative and mixes slowly.
In dimension 1, we show on the contrary that the dynamic setting has a fundamentally different nature: the speed of the walker is a monotonic function of the density of the SEP, with at most one critical value where it could vanish.
The proof uses a comparison with a family of finite-range models to handle the long-range correlations, and an original coupling to circumvent the poor mixing properties of the exclusion process.
This is joint work with Daniel Kious and Pierre-François Rodriguez.

In this talk, following the talk on Nov. 5, I will further review the asymptotic behavior of empirical spectral distribution (ESD) of the sparse random matrices. I will also present my recent work and some simulations.

In this talk, I will review the foundational framework for analyzing the convergence of large-dimensional random matrices' empirical spectral distribution (ESD) as matrix dimensions grow. If time permits, I will also present some simulation results and discuss my recent work.

A self-avoiding walk is a path on a lattice that does not visit the same site more than once. Despite this simple definition, many of the most basic questions are difficult to resolve in a mathematically rigorous fashion. However, since in high dimension the principal effect of the self-avoidance constraint become relatively weak, a lot of results have been proved, compared to low dimensional case. For example, introducing a parameter $z$, we consider a generating function $\chi_z=\sum_{n=0}^{\infty}C_nz^n$ where $C_n$ is the number of self-avoiding walks starting from origin. It is believed that there exists dimension-dependent exponent $\gamma$ such that $\chi_z$ diverges as $(z_c-z)^{-\gamma}$ when $z$ is getting close to the radius of convergence $z_c$.

Even the existence of such critical exponent is still unknown when $d\le4$, while it has been proved that $\gamma=1$ when $d>4$. The lace expansion is known as one of the most powerful tool to show the 'mean-field behavior' of statistical mechanical models in high dimension, which played an important role to prove $\gamma=1$ for $d>4$. In this seminar, we will explore how the lace expansion is derived and its mathematical foundation, and we will discuss how the lace expansion can be used to establish mean-field type results and specifically to prove that $\gamma=1$ for $d>4$.

A self-avoiding walk is a path on a lattice that does not visit the same site more than once. Despite this simple definition, many of the most basic questions are difficult to resolve in a mathematically rigorous fashion. However, since in high dimension the principal effect of the self-avoidance constraint become relatively weak, a lot of results have been proved, compared to low dimensional case. For example, introducing a parameter $z$, we consider a generating function $\chi_z=\sum_{n=0}^{\infty}C_nz^n$ where $C_n$ is the number of self-avoiding walks starting from origin. It is believed that there exists dimension-dependent exponent $\gamma$ such that $\chi_z$ diverges as $(z_c-z)^{-\gamma}$ when $z$ is getting close to the radius of convergence $z_c$.

Even the existence of such critical exponent is still unknown when $d\le4$, while it has been proved that $\gamma=1$ when $d>4$. The lace expansion is known as one of the most powerful tool to show the 'mean-field behavior' of statistical mechanical models in high dimension, which played an important role to prove $\gamma=1$ for $d>4$. In this seminar, we will explore how the lace expansion is derived and its mathematical foundation, and we will discuss how the lace expansion can be used to establish mean-field type results and specifically to prove that $\gamma=1$ for $d>4$.

A self-avoiding walk is a path on a lattice that does not visit the same site more than once. Despite this simple definition, many of the most basic questions are difficult to resolve in a mathematically rigorous fashion. However, since in high dimension the principal effect of the self-avoidance constraint become relatively weak, a lot of results have been proved, compared to low dimensional case. For example, introducing a parameter $z$, we consider a generating function $\chi_z=\sum_{n=0}^{\infty}C_nz^n$ where $C_n$ is the number of self-avoiding walks starting from origin. It is believed that there exists dimension-dependent exponent $\gamma$ such that $\chi_z$ diverges as $(z_c-z)^{-\gamma}$ when $z$ is getting close to the radius of convergence $z_c$.

Even the existence of such critical exponent is still unknown when $d\le4$, while it has been proved that $\gamma=1$ when $d>4$. The lace expansion is known as one of the most powerful tool to show the 'mean-field behavior' of statistical mechanical models in high dimension, which played an important role to prove $\gamma=1$ for $d>4$. In this seminar, we will explore how the lace expansion is derived and its mathematical foundation, and we will discuss how the lace expansion can be used to establish mean-field type results and specifically to prove that $\gamma=1$ for $d>4$.

The self-avoiding walk (SAW) is a model defined by adding self-avoidance interaction to the random walk. In other words, each path does not visit the same vertex on a graph more than once. We consider the connectivity function $c_n(x)$ defined by the number of $n$-step SAWs from the origin $o$ to a vertex $x$. It is known that the spread-out SAW with finite-range interactions enjoys the central limit theorem [van der Hofstad and Slade (2002) PTRF][van der Hofstad and Slade (2003) AAM]. Taking an average on a ball, they also proved a certain type of a local limit theorem for $c_n(x)$. For the spread-out SAW with long-range interactions whose one-step distribution has heavy tails, the power-law decay of the two-point function $G_p(x) = \sum_{x \in \mathbb{Z}^d} c_n(x) p^n$ was shown in [Chen and Sakai (2015) AOP][Chen and Sakai (2019) CMP].

In this talk, I will explain an attempt to prove a local limit theorem for the spread-out long-range SAW in the original sense. Our motivations come from combining the results of the previous researches. I will show two different strategies. The lace expansion gives a certain type of a recurrence relation for the sequence $\{c_n(x)\}_{n=1}^{\infty}$. The first one is based on the analogous approach with [Chen and Sakai (2019) CMP] in which we substitute the recurrence relation into $\{c_i(x)\}_{i=1}^{n-1}$. The second one is based on the inductive approach [van der Hofstad and Slade (2002) PTRF] extended to the long-range model in which we assume an upper bound on $c_n(x)$ for $1 \leq m \leq n$ and prove it for $2 \leq m \leq n+1$. I will report the current progress of our attempts using these approaches.

This talk is joint work with Lung-Chi Chen (National Chengchi University) and Yuki Chino (National Yang-Ming Chiao-Tung University).

In this lecture, we will review recent works regarding spectral statistics of the normalized adjacency matrices of random $d$-regular graphs on $N$ vertices. Denote their eigenvalues by $\lambda_1 = d/\sqrt{d-1} \geq \lambda_2 \geq \dots \geq \lambda_N$, and let $\gamma_i$ be the classical location of the $i$-th eigenvalue under the Kesten-McKay law. Our main result asserts that for any $d \geq 3$ the optimal eigenvalue rigidity holds in the sense that $$|\lambda_i − \gamma_i| \leq \frac{N^{o_N(1)}}{N^{2/3}(\min \{ i, N − i + 1 \})^{1/3} }, \quad \forall i \in \{ 2, 3, \dots, N \}$$ with probability $1 − N^{−1+o_N(1)}$. In particular, the characteristic $N^{−2/3}$ fluctuations for Tracy-Widom law is established for the second largest eigenvalue. Furthermore, for $d \geq N \epsilon$ for any $\epsilon > 0$ fixed, the extremal eigenvalues obey the Tracy-Widom law. This is a joint work with Jiaoyang Huang and Theo McKenzie.

In Bernoulli percolation, the incipient infinite cluster (IIC) is a version of the "open cluster of the origin at criticality, conditioned to be infinite". Since this event should have probability 0 on $\mathbb{Z}^d$, the IIC is constructed via a limiting procedure. For $d > 6$, several constructions have been given and shown to produce the same object, but many natural limiting procedures remain unexplored. For instance, it is an open question whether conditioning on $\{\text{$0$ is connected to $\partial [-n, n]^d$}\}$ produces the IIC as $n \to \infty$. We answer this question in the affirmative as a corollary of our theorem, which roughly says "conditioning on any long open connection produces the IIC", and whose proof does not directly use lace expansion analysis.

Understanding when a high-dimensional polytope contains integer points is a fundamental problem in a wide variety of fields including combinatorial optimization, computer science, Banach geometry, statistical physics, and information theory. We introduce a general random model of this problem that encodes: the closest vector problem, linear feasibility, integer linear feasibility, perceptron problems, and combinatorial discrepancy in any norm. We study the "margin", the distance between a polytope and the nearest integer points. The margin acts as a quantitative measure for the "distance to feasibility" for random optimization problems. Our main result is a set of sufficient conditions for the margin to concentrate. Concentration of the margin implies a host of new sharp threshold results in the mentioned models, and also simplifies and extends some key known results.

References:
[1] https://arxiv.org/abs/2309.13133
[2] https://arxiv.org/abs/2205.02319

In this talk, we will derive the Stein equations for the Poisson and gamma distributions, and utilize them to estimate the errors of the Poisson, exponential, and Chi-square approximations. In addition to the integration by parts method, we will also introduce another approach involving the utilization of the generator of a diffusion process to derive the Stein equation for its stationary distribution.

In this talk, I will introduce the multidimensional Stein method. The content will cover: (1) the Gaussian integration by parts formula, (2) the Stein equation for normal random vectors along with the properties of its solution, and (3) the Gaussian Poincaré inequality.

In this work, we study the critical long-range percolation on $\mathbb{Z}$, where a long-range edge connects $i$ and $j$ independently with probability $\beta|i-j|^{-2}$ for some fixed $\beta > 0$. Viewing this as a random electric network where each edge has a unit conductance, we show that with high probability the effective resistance from the origin $0$ to $[-N, N]^c$ has a polynomial lower bound in $N$. Our bound holds for any $\beta > 0$ and thus rules out a potential phase transition (around $\beta = 1$) which seemed to be a reasonable possibility. This is a joint work with Jian Ding and Zherui Fan.

The main goal of this lecture series is to introduce some emerging topics, such as phase transitions and algorithmic aspects, for the number partitioning problems and perceptron models. These are fundamental models arising from computer science and neural network and known to exhibit profound structures that are mathematically challenging to study and numerically difficult to simulate. This lecture series will be a great opportunity for the young scholars including undergraduate and graduate students and junior researchers to learn and expand their research interests.

Probability theory is foundational in understanding random phenomena, with the central limit theorem being a cornerstone result. This theorem establishes that the standardized partial sums of independent and identically distributed random variables converge in distribution to a Gaussian random variable. In the 1970s, Charles Stein pioneered a method for proving generalized central limit theorems, with relaxed assumptions to allow for more application scenarios. More importantly, this method allows us to estimate the distance between two probability distributions using differential operators. Since then, many authors in literature have obtained quantitative central limit theorems and found applications in different areas using the so-called Stein’s method. In this talk, we will introduce Stein's method and its applications.

In this talk, we will first introduce the motivation of studying stochastic Navier-Stokes equations with Markov switching. Then we will sketch the procedure of solving such an equation. If time permits, we will mention more technicalities.

In this talk, we report a quite unexpected connection of two legendary topics, Normal Numbers (Emile Borel, 1909) and Brownian Motions (Albert Einstein, 1905; Norbert Wiener 1929, and Paul Levy 1930’s). The "bridge" of the connection is Hermann Weyl's pioneering exponential sum work (1919) and its later investigations by various authors. The talk is based on the speaker's article appeared in Mathematical Research Reports, vol. 2, 2021. The article is a memory of the late Professor Samuel James Taylor, and the math techniques are cited from a famous 1985 book “Some Random Series of Functions” by the late Professor Jean-Pierre Kahane.

In applications the underlying probability may be known up to a normalizing constant only, the direct sampling is not possible. Instead, Markov processes could be used for approximations. In this talk I'll describe some mathematical setups and related problems motivated by this application.

In this lecture, we'll review some recent results regarding spectral gaps and logarithmic Sobolev inequality for Glauber dynamics of mean-field spin glass models. In particular, we will present a method to prove that the spectral gap of the Glauber dynamics is of order one at sufficiently high temperature. In addition, we will review certain estimates on two point functions for the SK model satisfying a modified AT condition.

Within the province of condensed matter physics, there exists a variety of interesting physical phenomena where we are concerned with the statistical fluctuations exhibited by an essentially linear elastic object, such as a hydrophilic polymer chain wafting in water. Due to the thermal fluctuation, the shape of the polymer chain should be understood as a random path. The water in this physical system plays the role of the disordered environment that contains randomly placed hydrophobic molecules as impurities, which repel the hydrophilic monomers that the polymer chain consists of. This physical system is called the directed polymer model. The major problem about this system is to investigate the behavior of the polymer chain for various disorder strengths.

In this talk, we will focus on the $(d+1)$-dimensional directed polymer model when $d \geq 3$. In this case, the system is quite sensitive to the disorder strength. Moreover, it is widely believed that the model has a phase transition when the disorder strength is strong enough. Consequently, it is crucial to investigate the behavior of system until a critical disorder strength that the system may have a phase transition. As a result, the goal of this talk is to present the results given by D. Lygkonis and N. Zygouras [1], which studied the limiting fluctuations of the partition function and the free energy of the directed polymer model when the disorder strength is less than a critical value.

[1] D. Lygkonis and N. Zygouras. Edwards–Wilkinson fluctuations for the directed polymer in the full $L^2$-regime for dimensions $d \geq 3$. Annales de l’Institut Henri Poincaré, Probabilités et Statistiques, 58(1):65 – 104, 2022. doi:10.1214/21-AIHP1173.

Within the province of condensed matter physics, there exists a variety of interesting physical phenomena where we are concerned with the statistical fluctuations exhibited by an essentially linear elastic object, such as a hydrophilic polymer chain wafting in water. Due to the thermal fluctuation, the shape of the polymer chain should be understood as a random path. The water in this physical system plays the role of the disordered environment that contains randomly placed hydrophobic molecules as impurities, which repel the hydrophilic monomers that the polymer chain consists of. This physical system is called the directed polymer model. The major problem about this system is to investigate the behavior of the polymer chain for various disorder strengths.

In this talk, we will focus on the $(d+1)$-dimensional directed polymer model when $d \geq 3$. In this case, the system is quite sensitive to the disorder strength. Moreover, it is widely believed that the model has a phase transition when the disorder strength is strong enough. Consequently, it is crucial to investigate the behavior of system until a critical disorder strength that the system may have a phase transition. As a result, the goal of this talk is to present the results given by D. Lygkonis and N. Zygouras [1], which studied the limiting fluctuations of the partition function and the free energy of the directed polymer model when the disorder strength is less than a critical value.

[1] D. Lygkonis and N. Zygouras. Edwards–Wilkinson fluctuations for the directed polymer in the full $L^2$-regime for dimensions $d \geq 3$. Annales de l’Institut Henri Poincaré, Probabilités et Statistiques, 58(1):65 – 104, 2022. doi:10.1214/21-AIHP1173.

The famous Gaussian Unitary Ensemble (GUE) consists of random Hermitian matrices with elements identically and independently (up to symmetry) distributed as Gaussian random variables. It also has the property that its probability measure is invariant under unitary conjugations. In general unitary invariant ensembles, one maintains the latter property but generalizes the measure by including a potential. We explain how the correlation functions of eigenvalues for such an ensemble can be compactly represented in terms of orthogonal polynomials for a weight involving the potential. Then we demonstrate how Deift, Kriecherbauer, McLaughlin, Venakides, and Zhou proved universality of the correlation functions both in the bulk of the spectrum and at the edge for general convex and analytic potentials, vastly generalizing known results for GUE. Their method uses a characterization of orthogonal polynomials in terms of the solution of a matrix Riemann-Hilbert problem, which applies for completely general weights. Finally, we discuss how the methodology was further extended by McLaughlin and the speaker to extend the universality results to potentials having just two Lipschitz continuous derivatives.

Phenomena of non-equilibrium growth processes are ubiquitous in nature. Many of these processes are believed to exhibit remarkably similar growth dynamics, which can be regarded as an interface evolves with time, changing its roughness while being subjected to random noise. In the original 1986 paper of M. Kardar, G. Parisi and Y.-C. Zhang [2], the authors predicted the dynamic of these processes, and indicated that the evolution can be described by the solution $\mathscr{H}(x,t)$ of a stochastic PDE, which is called the Kardar-Parisi-Zhang (KPZ) equation. On the other hand, $\mathscr{H}(x,t)$ plays an important role in statistical mechanics. Roughly speaking, through the Cole–Hopf transform $\mathscr{H}(x,t) = \log \mathscr{U}(x,t)$, $\mathscr{H}(x,t)$ can be regarded as the free energy of the continuous directed polymer model, which describes the behaviour of a hydrophilic polymer chain wafting in a disordered environment that contains randomly placed hydrophobic molecules as impurities. Here $\mathscr{U}(x,t)$ plays the role of the partition function of this model, and solves a stochastic PDE, which is called the stochastic heat equation (SHE).

In this talk, we will focus on the SHE and KPZ equation when spatial dimension $d\geq 3$. In this case, both of the SHE and KPZ equation are quite sensitive to $\beta$, where $\beta$ describes the strength of the white noise in these equations. Consequently, it is crucial to investigate the behavior of the SHE and KPZ equation for all $\beta$. In a recent paper of F. Comets, C. Cosco, and C. Mukherjee [1], the authors proved the limiting fluctuation of $\mathscr{H}(x,t)$ under a restriction on $\beta$. Motivated by this result, in our recent work [3], we considered both $\mathscr{H}(x,t)$ and $\mathscr{U}(x,t)$, and established their limiting fluctuations in the entire $L^{2}$-regime (i.e., $\beta < \beta_{L^{2}}$). Here $\beta_{L^{2}}$ is a critical value associated with the KPZ equation.

[1] F. Comets, C. Cosco, and C. Mukherjee. Space-time fluctuation of the Kardar-Parisi-Zhang equation in d $\geq 3$ and the Gaussian free field. arXiv preprint, 2019. doi:10.48550/arXiv.1905.03200.

[2] M. Kardar, G. Parisi, and Y.-C. Zhang. Dynamic scaling of growing interfaces. Phys. Rev. Lett., 56:889–892, 1986. doi:10.1103/PhysRevLett.56.889.

[3] T.-C. Wang. Space-time fluctuations for the SHE and KPZ equation in the entire $L^2$-regime for spatial dimensions $d \geq 3$. in preparation. 2023+.

For a large class of second order elliptic differential operator $\mathcal{L}$, there is a related diffusion process X in which the operator $\mathcal{L}$ becomes an infinitesimal generator of X. The fundamental solution of heat equation of $\mathcal{L}$ is the transition density of the process X. In this talk, we will discuss theses relation of the general Lévy process through the Lévy Khintchine formula and its transition density.

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

Stochastic processes are mathematical models of random phenomena and Lévy process is a large class of stochastic process. This course is an introduction to the theory of Lévy processes that covers definitions, infinitely divisibility and characteristic exponents. We also discuss examples of Lévy processes and Lévy Khintchine formula.

We consider the spread-out models of the self-avoiding walk and its finite-memory version, called the memory-$\tau$ walk. For both models, each step is uniformly distributed over the d-dimensional box $\{ x \in \mathbb{Z}^d : 0 < |x|_{\infty} \le L \}$. The critical point $p_c^{\tau}$ for the memory-$\tau$ walk is increasing in $\tau$ and converges to the critical point for the self-avoiding walk as $\tau$ goes to $\infty$. The speaker proved that the rate of convergence of $p_c^{\tau}$ in terms of $\tau$ is order of $\tau^{-(d-2)/2}$. Moreover, the speaker identified the exact expression of the coefficient of the dominant term of it. This improves the previous results obtained by Madras and Slade [Birkhäuser, The Self-Avoiding Walk, Lemma 6.8.6, 1993]. This talk is based on the speaker’s own work (http://arxiv.org/abs/2306.13936).

Stochastic processes are mathematical models of random phenomena and Lévy process is a large class of stochastic process. This course is an introduction to the theory of Lévy processes that covers definitions, infinitely divisibility and characteristic exponents. We also discuss examples of Lévy processes and Lévy Khintchine formula.

The quantum Ising model is a kind of model of ferromagnetic materials. According to quantum mechanics, ferromagnetism comes from a cooperation phenomenon of spins. The Hamiltonian operator (energy) for the quantum Ising model is defined by tensor products of Pauli matrices, which correspond to spins as a physical quantity. In this model, we impose a transverse field. The case without the transverse field is called the classical Ising model in particular. It is predicted that changing temperature or a transverse field causes phase transitions and critical phenomena from a point of view of numerical analysis (though this fact is mathematically proven in some special cases). It is well-known that critical exponents for the classical Ising model take the mean-field values in high dimensions. We are interested in whether or not the values are changed even when we impose the transverse field. In this series of talks, I will focus on the critical exponent $\gamma$ of the magnetic susceptibility and show that $\gamma = 1$ still holds for the quantum Ising model with the nearest-neighbor interaction. I will also mention an ongoing work about the lace expansion, which plays an important role in proving mean-field behavior for a lot of statistical mechanical models in high dimensions.

Directed Random Polymers model the motion of a random walk in a random potential. From its introduction into the mathematical community 35 years ago it has offered the ground for many interesting mathematical developments. In this talk I will review some of the milestones and point to some of the still tantalising open questions.

The quantum Ising model is a kind of model of ferromagnetic materials. According to quantum mechanics, ferromagnetism comes from a cooperation phenomenon of spins. The Hamiltonian operator (energy) for the quantum Ising model is defined by tensor products of Pauli matrices, which correspond to spins as a physical quantity. In this model, we impose a transverse field. The case without the transverse field is called the classical Ising model in particular. It is predicted that changing temperature or a transverse field causes phase transitions and critical phenomena from a point of view of numerical analysis (though this fact is mathematically proven in some special cases). It is well-known that critical exponents for the classical Ising model take the mean-field values in high dimensions. We are interested in whether or not the values are changed even when we impose the transverse field. In this series of talks, I will focus on the critical exponent $\gamma$ of the magnetic susceptibility and show that $\gamma = 1$ still holds for the quantum Ising model with the nearest-neighbor interaction. I will also mention an ongoing work about the lace expansion, which plays an important role in proving mean-field behavior for a lot of statistical mechanical models in high dimensions.

Is it true that if many elements of a group commute, then the group is commutative? What happens if most of the elements have order 2? Can every group be realised as the automorphism group of a "nice" graph? These questions have been solved in the 70-90's for finite groups. But what happens for infinite groups? How can we interpret "many elements" or "most of the elements" for infinite groups?

In this talk we will see how we can use random walks to solve the above questions for infinite (finitely generated) groups.

The talk is supposed to be a gentle introduction to random walks and infinite group theory; showing how probability theory and algebra nicely interact. No prerequisites are required and the talk will accessible to second year bachelor students.

In this talk, our lecture starts with introducing pre-Brownian motion which is defined from a Gaussian white noise. Going from pre-Brownian motion to Brownian motion requires the additional property of continuity of sample paths. Then we introduce some properties of Brownian sample paths, and establishes the strong Markov property.

The quantum Ising model is a kind of model of ferromagnetic materials. According to quantum mechanics, ferromagnetism comes from a cooperation phenomenon of spins. The Hamiltonian operator (energy) for the quantum Ising model is defined by tensor products of Pauli matrices, which correspond to spins as a physical quantity. In this model, we impose a transverse field. The case without the transverse field is called the classical Ising model in particular. It is predicted that changing temperature or a transverse field causes phase transitions and critical phenomena from a point of view of numerical analysis (though this fact is mathematically proven in some special cases). It is well-known that critical exponents for the classical Ising model take the mean-field values in high dimensions. We are interested in whether or not the values are changed even when we impose the transverse field. In this series of talks, I will focus on the critical exponent $\gamma$ of the magnetic susceptibility and show that $\gamma = 1$ still holds for the quantum Ising model with the nearest-neighbor interaction. I will also mention an ongoing work about the lace expansion, which plays an important role in proving mean-field behavior for a lot of statistical mechanical models in high dimensions.

In the first lecture we discussed extreme value theory for logarithmically correlated Gaussian fields. In this talk I will discuss what changes in the non-Gaussian Gaussian setup. A prime example is the study of cover time of certain planar graphs or two dimensional manifolds by random walk or Brownian motion. In spite of precise and beautiful links through isomorphism theorems, the question about the cover time of the 2D torus by a Wiener sausage (or its discrete analogue) requires new tools. I will describe some work, old and recent, on this question, culminating with limit theorems for the cover time. If time permits, I will briefly discuss other types of non Gaussian LCFs.

The discrete Gaussian free field (DGFF) and the simple random walk (SRW) have a close relationship via the generalized second Ray-Knight theorem, which is a distributional identity between the square of DGFF and the local time of SRW. Thanks to the theorem, we have witnessed rapid progress on the studies of the cover time (the first time at which SRW visits all the vertices) and thick points of SRW (sites frequently visited by SRW). In the first half of this lecture, I will state the generalized second Ray-Knight theorem and review results on the cover time due to Ding-Lee-Peres (2012) and Zhai (2018) where we can see beautiful applications of the theorem. In the second half, I will focus on applications to thick points of SRW.

The extreme value theory for Gaussian logarithmically correlated fields (G-LCFs) has emerged in the last decade as a powerful tool in the analysis of interface models, quantum gravity, random matrices and in a myriad of other applications.
The two dimensional Gaussian free field (and its discrete analogue) is an important motivating example of such a field. In this lecture, I will describe the relation and differences between the extreme value theory for i.i.d. variables and that for G.-LCFs, and introduce the relation with branching structures and various tools such as comparison theorems, scale decompositions and relations to branching random walks.

The discrete Gaussian free field (DGFF) is a centered Gaussian field on a graph whose covariance is given by the inverse of the graph Laplacian. It is a probabilistic model of interfaces and has connections with a lot of other models such as local times of random walks and branching random walks. In the first half of this lecture, I will give some motivation and basics of DGFF such as the random walk representation and the domain Markov property. In the second half, I will review some progress on the extreme value theory of DGFF on the integer lattice in three or higher dimensions.

Self-avoiding walk (SAW) is a walk that each path does not visit the same point more than once. It is a model of fundamental interest in combinatorics, probability theory, statistical physics and polymer chemistry. It is certainly non-Markovian. These features makes the subject difficult, and many of the central problems remain unsolved.
This is the second talk of the series of lectures. In this talk we focus on the diagrammatic estimates of the lace expansion for SAW, and the random-walk estimates.

Self-avoiding walk (SAW) is a walk that each path does not visit the same point more than once. It is a model of fundamental interest in combinatorics, probability theory, statistical physics and polymer chemistry. It is certainly non-Markovian. These features makes the subject difficult, and many of the central problems remain unsolved.
This is the last talk of the series of lectures. In this talk, we prove convergence of the lace expansion for the nearest-neighbor model in sufficiently high dimension, and for sufficiently spread-out model in dimensions d > 4. Furthermore, we discuss further results for SAW in high dimensions.