Welcome to the XXIV Brazilian School of Probability which will take place from August 31st to September 4th, 2020. Due to the COVID19 pandemic the school will be online this year. There are two ways to attend the school, the first is via the platform Google meet which has a limitation on the amount of participants. Each day about 30 minutes before the first course we will make the link available on the bottom of this page. The second way is via the youtube channel of impa https://www.youtube.com/impabr, which also has a chat function and which allows to ask questions which then either will be answered by a moderator or will be forwarded to the speaker. We recommend attending the school via youtube for those who have an unstable internet connection.

HERE you can find the recordings of the meetings

## Perla Sousi

( University of Cambridge)**Title:** Random walks and uniform spanning trees

**Abstract:** Spanning trees in a connected graph are basic objects of great interest in combinatorics and computer science. In this course we will study what a typical spanning tree looks like and we will discuss elegant sampling algorithms using random walks due to Aldous, Broder and Wilson from the 1990's. We will then explain how we can sample a uniform spanning tree (or forest) of $\mathbb{Z}^d$ for $d\geq 2$. Using the connection with random walks and a recent algorithm due to Hutchcroft we will obtain results on the fine geometry of uniform spanning trees.

## Martin Hairer

(Imperial College London)**Title:** The Brownian Castle

**Abstract:** We will start with an overview of Wilson's RG picture for 1+1 dimensional dynamics and discuss the EW and KPZ fixed points. We then show that there is another non-trivial RG fixed point (which is "free" but not Gaussian) which we call the Brownian Castle. The minicourse will be devoted to the proof of some of its pathwise properties, including a new construction of the Brownian web (in a stronger space than usual) which arises as the main ingredient in its construction.