Accompanying workshops will be held on 25th May 2026 from 11 am in room 1016.
11:00 - Mikołaj Frączyk: Percolations and phase transitions for toddlers
12:15 - Michał Kotowski: Cycle structure of random permutations and related loop models
13:15 - Break
14:30 - Minghao Pan: Branching random walks and percolation in high dimensions
Abstracts:
Michał Kotowski (UW)
Title: Cycle structure of random permutations and related loop models
In the talk I will give a survey of the cycle structure of random permutations evolving in time. The prototypical example is the so-called interchange process on finite graphs in which permutations evolve by random transpositions. The main question, much in the same spirit as in percolation theory, is the following: how quickly do macroscopic cycles emerge in such a process? There is also a counterpart for infinite graphs, related to a well-known conjecture due to Bálint Tóth on the interchange process on Z^d. If time permits, I will discuss the relation to the Heisenberg model from statistical physics and other loop models.
Mikołaj Frączyk (UJ)
Title: Percolations and phase transitions for toddlers
I will define Bernoulli bond and site percolations and introduce critical percolation thresholds p_c and p_u. Then we shall run a few simulations involving different statistical models to illustrate what a phase transition is. The talk will contain no proofs and should be accessible to non-experts.
Minghao Pan (California Institute of Technology)
Title: Branching random walks and percolation in high dimensions
This introductory talk explores the connection between branching random walks and percolation as models of random growth and connectivity. We will explain how percolation clusters in high dimensions often resemble branching processes, leading to “mean-field” behavior near criticality. The talk will focus on basic definitions, key phenomena such as survival and cluster size, and the intuition behind why high-dimensional geometry makes branching approximations effective.