台大校區機率演講公告Probability Seminars on NTU Campus

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.

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.