15:30 - 16:30

Analyzing Spectral Convergence in Large-Dimensional Random Matrices: Part 2

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.

15:30 - 16:30

Analyzing Spectral Convergence in Large-Dimensional Random Matrices: Part 1

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.

15:30 - 17:00

The lace expansion for the self-avoiding walk (Part 3)

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 $d>4$.

15:30 - 17:00

The lace expansion for the self-avoiding walk (Part 2)

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 $d>4$.

15:30 - 16:30

The lace expansion for the self-avoiding walk (Part 1)

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 $d>4$.

15:30 - 16:30

Attempts to prove a local limit theorem for the long-range self-avoiding walk

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).

14:00 - 15:30

Spectral Statistics of Random Regular Graphs

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.

15:30 - 16:30

Robust construction of the high-dimensional incipient infinite cluster

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.

10:00 - 11:00

Zero-one Laws for Random Feasibility Problems

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

15:30 - 16:30

Introduction to Stein's Method: Part III

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.

15:30 - 16:30

Introduction to Stein's Method: Part II

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.

15:30 - 16:30

Polynomial lower bound on the effective resistance for the one-dimensional critical long-range percolation

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.

15:30 - 16:45

15:30 - 16:45

10:00 - 11:15

15:30 - 16:45

10:00 - 11:15

Phase Transitions and Algorithmic Hardness for the Number Partitioning Problems and Perceptron Models

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.

15:30 - 16:30

Introduction to Stein's Method: Part I

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.

14:00 - 15:00

Three-dimensional stochastic Navier-Stokes equations with Markov switching

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.

15:30 - 16:30

A Tale of Two Legacies

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.

15:30 - 16:30

On Dynamic Monte Carlo / Markov Chain Monte Carlo

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.

14:00 - 15:00

Spectral Gap and Two Point Function Estimates for Mean-field Spin Glass Models

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.

14:00 - 15:00

On the Gaussian asymptotics of the $(d+1)$-dimensional directed polymer model in the entire $L^{2}$-regime for dimensions $d \geq 3$: Part II

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.

10:30 - 11:30

On the Gaussian asymptotics of the $(d+1)$-dimensional directed polymer model in the entire $L^{2}$-regime for dimensions $d \geq 3$: Part I

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.

15:30 - 16:30

Universality in Random Matrix Theory

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.

10:30 - 11:30

On the Gaussian asymptotics of the SHE and KPZ equation in the entire $L^{2}$-regime for spatial dimensions $d \geq 3$

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+.

15:30 - 17:00

Introduction to Lévy process: Part III

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.

16:10 - 17:30

Universality of log-correlated fields

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.

15:30 - 17:00

Introduction to Lévy process: Part II

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.

15:30 - 16:30

Rate of convergence of the critical point for the memory-$\tau$ self-avoiding walk in dimension $d>4$

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).

15:30 - 17:00

Introduction to Lévy process: Part I

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.

15:30 - 17:00

Stability of the phase transition and critical behavior of the Ising model against quantum perturbation: Part III

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.

14:00 - 15:00

35 years of Directed Random Polymers

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.

15:30 - 17:00

Stability of the phase transition and critical behavior of the Ising model against quantum perturbation: Part II

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.

14:00 - 15:00

Random walks on infinite groups: some applications

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.

13:30 - 17:00

Introduction: Some Properties of Continuity of Sample Paths for Brownian Motion

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.

15:30 - 17:00

Stability of the phase transition and critical behavior of the Ising model against quantum perturbation: Part I

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.

15:30 - 17:30

Extreme value theory for non-Gaussian logarithmically correlated fields

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.

13:30 - 15:10

Applications of the discrete Gaussian free field to random walks

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.

15:30 - 17:30

Extreme value theory for Gaussian logarithmically correlated fields

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.

13:30 - 15:10

Introduction to the discrete Gaussian free field

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.