Baining Zhang’s Homepage
Hello, I am a third year undergraduate student at HKUST, currently on exchange at UBC Vancouver. I run a more mathematical blog called Peasant Maths.
Academic Interests
I am mainly interested in probability theory and related fields, including mathematical physics and the geometry of discrete groups. Most recently, I have been studying random walks (in random environments).
An idea which I would like to explore is the dream of universality, which (vaguely) says that the behaviour of a large system is independent of the microscopic details. A first example of universality is the central limit theorem: the rescaled distribution of the mean of a large number of random variables behaves Gaussian. CLT-type results are still very much the goals when proving universality results for other types of stochastic objects. To physicists, universality is a fundamental underlying principle of statistical mechanics. Yet, for mathematicians, a rigorous understanding of universal phenomena and justification for the principle of universality is still very much a dream. This excellent talk by Martin Hairer should explain one tip of the iceberg.
Notes
Towards Schramm-Loewner Evolutions
This set of notes was completed during a reading course titled Conformally Invariant Processes in the Plane. The part on univalent functions is not very well-written. A highlight of the notes is a partial proof of the measurability of the Loewner map, which takes a Brownian motion path to an SLE-path.
A list of references which I found helpful is included at the end of the notes.
Loop-Erased Random Walks
This topic was learned as a part of my SCIE 2500 Guided Study on Research II project. The reports and presentation can be found below. Note however that due to length limitations, they only provide glimpses of an introduction to LERWs.
Progress Report
Final Report
Presentation Slides
Lawler has a book Random Explorations on this topic. He also has a set of notes titled Topics in loop measures and the loop-erased walk, which has many typos. I recommend reading Random Explorations first.
Random Walks and Homogenization Theory
Handwritten notes taken from lecture videos (in Chinese) of a course by Chenlin Gu.
Course Webpage
Lecture 1
Lecture 2
Lecture 3
Lecture 4
Lecture 5
Certainly contain a lot of mistakes.
Update: I was scammed. I thought a complete proof of the invariance principle for the random conductance model would be given in the first 5 weeks, as suggested by the course schedule. But it really does seem like I need PDE knowledge to understand the corrector. I will return to this after a course on PDEs.
Self-Avoiding Walks in Random Environments
The following report was written in the context of an undergraduate ‘research’ project on self-avoiding walks in random conductances and supercritical percolation clusters. We slightly modify the setting of the problem in a paper by Chino and Sakai and essentially show that the same proof still works in the new setting.
The Quenched Critical Point of Self-Avoiding Walk on Infinite Graphs with Random Conductances