报告题目：Littlewood-Offord bounds on the symmetric groups and applications

报告人：Tuan Tran 教授

报告时间：2026年1月9日10:00-11:00

报告地点：科技园阳光楼南815

邀请人：高国荣

邀请单位：福州大学数学与统计学院

报告内容简介：

The anti-concentration phenomenon in probability theory has been intensively studied in recent years, with applications across many areas of mathematics. In most existing works, the ambient probability space is a product space generated by independent random variables.

In this work, we initiate a systematic study of anti-concentration when the ambient space is the symmetric group, equipped with the uniform measure.Concretely, we focus on the random sum

\[S_{\pi} := \sum_{i=1}^n v_i w_{\pi(i)},\]

where $\mathbf{v} = (v_1,\dots,v_n)$ and $\mathbf{w} = (w_1,\dots,w_n)$ are fixed vectors and $\pi$ is a uniformly random permutation of $[n]$.

This work contains several new results, addressing both discrete and continuous anti-concentration phenomena. On the discrete side, we establish a near-optimal structural characterization of the vectors $\mathbf{v}$ and $\mathbf{w}$ under the assumption that the concentration probability

\[\sup_{x \in \mathbb{R}} \mathbb{P}(S_{\pi} = x)\]

is polynomially large. As applications, we derive and strengthen a number of previous results. In particular, we show that if both $\mathbf{v}$ and $\mathbf{w}$ have distinct entries, then

\[\sup_{x \in \mathbb{R}} \mathbb{P}(S_{\pi} = x) = O(n^{-1}).\]

This bound serves as a permutation-space analogue of the classical Erdős-Moser bound in the product-space setting and answers a question posed by Alon-Pohoata-Zhu.

From the continuous perspective, we study the small-ball event

\[\mathbb{P}(|S_{\pi} - x| \le \varepsilon).\]

We establish sharp bounds in various settings, including results exhibiting sub-Gaussian decay in $\varepsilon$. With additional effort, we are also able to treat the joint distribution of these events. Moreover, we provide a characterization of the vectors $\mathbf{v}$ and $\mathbf{w}$ for which these small-ball probabilities are large. As an application, we prove that the number of extremal points of random permutation polynomials is bounded by $O(n^{2/3})$, extending results of S{\"o}ze on the number of real roots.

报告人简介：

Tuan Tran为中国科学技术大学教授，获得海外优秀青年基金资助。Tuan Tran教授曾担任过乌姆相一主持的基础科学研究所青年科学家研究员、河内科技大学讲师、苏黎世联邦理工学院Feodor Lynen研究员、由Benny Sudakov主持，以及由Diana Piguet主持的捷克科学院ICS博士后。2015年在Tibor Szabó指导下获得柏林自由大学博士学位。Tuan Tran教授在 JEMS、IJM、IMRN、JCTB等国际顶级期刊上发表了许多优秀的工作。