Hello,
I am a third year undergraduate student at HKUST. My interests centre around probability theory and mathematical physics. Most recently, I have been studying directed polymers, random interlacements, and the Brownian map.
Email: bzhangbp at connect dot ust dot hk
Notes
MATH 5380 (Topics in) Combinatorics
Note which I have taken from the lectures. Like the lectures, these notes view mathematics through an expressionist lens: they are not designed to make complete sense, but are meant to evoke thought.
Lecture 1 - 2026-02-04
Lecture 2 - 2026-02-11
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
Towards Schramm-Loewner Evolutions
This set of notes was completed during a reading course titled Conformally Invariant Processes in the Plane. The part on capacity and half-plane capacity is poorly written, but the proofs should be correct.
An extensive list of references and learning material which I found helpful is included at the end of the notes. In fact this is probably the most helpful part 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. For further content (especially on the uniform spanning forest), I recommend Probability on Trees and Networks by Lyons and Peres. Or just read that book anyway; it is basically the most beautiful book ever written.
Random Walks and Homogenization Theory
Incomplete poorly handwritten notes taken from lecture videos of a course by Chenlin Gu. Contains many mistakes which I do not intend to fix. These are available here mainly for my ease of access.
The lectures are in Chinese, but my notes are in English.
Course Webpage (with lecture videos)
Note 1
Note 2
Note 3
Note 4
Note 5
I will return to this after a course on PDEs.