报告题目: Cover times for random walks and extreme values of Gaussian free fields
报告人：Ding Jian (丁剑)，University of Chicago助理教授，湖南常德人。他2006年本科毕业于北京大学，2011年博士毕业于UC Berkele，导师为Yuval Peres教授（Fellow of the American Mathematical Society ）。他目前已经发表高质量的学术论文30多篇，其中在全世界最顶尖的数学期刊《Annals of Mathematics》和《Acta Mathematica》上各发表论文1篇，在概率领域里最顶尖的学术期刊《Annals of Probability》和《Probability Theory and Related Fields》上各发表论文5篇和2篇。其主要研究兴趣：Probability theory with focus on interactions with statistical physics and theory of computer science. In particular, extreme values of Gaussian processes, random constraint satisfaction problems, random planar geometry.
报告摘要： I will give a review talk on cover times for random walks, with
emphasis on the connection to the maxima of Gaussian free fields. The talk
will be an overview and will be of little technical details. The purpose
of the talk is to describe the following *story*.
In 2011, in a work of James Lee, Yuval Peres and myself, we estimated the
cover time for an arbitrary graph up to an absolute multiplicative
constant using the Ray-Knight isomorphism theorem (which links random walk
to Gaussian free fields) as well as Talagrand's majorizing measure
theory. This yields a deterministic polynomial time algorithm that
approximates the cover time up to constant, solving a question of Aldous
and Fill (1996).
In 2012, I improved the previous result by obtaining the asymptotics on
bounded-degree graphs as well as trees, using a connection with the random
current model as well as a coupling between random walks and Gaussian free
fields (on trees).
In 2015, Alex Zhai (and independent Titus Lupu) obtained the
aforementioned coupling between random walk and Gaussian free field for
general graphs, which yields a sharp estimate (together with exponential
concentration) for cover times.