Tag - Percolation theory

Jason Miller: Conformal Removability of SLE

6th October, 2023

We consider the Schramm-Loewner evolution (SLEκ) with κ = 4, the critical value of κ > 0 at or below which SLEκ is a simple curve and above which it is self-intersecting. We show that the range of an SLE4 curve is a.s. conformally removable. Such curves arise as the conformal welding of a pair of independent critical (γ = 2) Liouville quantum gravity (LQG) surfaces along their boundaries and our result implies that this conformal welding is unique. In order to establish this result, we give a new sufficient condition for a set X ⊆ ℂ to be conformally removable which applies in the case that X is not necessarily the boundary of a simply connected domain. We will also describe how this theorem can be applied to obtain the conformal removability of the SLEκ curves for κ ∈ (4,8) in the case that the adjacency graph of connected components of the complement is a.s. connected. This talk will assume no prior knowledge of SLE or LQG.

Matthew Tointon: Percolation on finite transitive graphs

2nd March, 2022

In Bernoulli bond percolation, one defines a random subgraph of a given connected graph G by deleting or retaining edges of G independently at random, each edge being retained with the same probability p. When G is infinite, a central and classical question is whether there exists a choice of p strictly less than 1 such that this random subgraph has infinite connected components. When G is finite, a natural analogue is to ask whether there exists a choice of p bounded away from 1 such that the random subgraph contains a connected component containing at least half (or 1% or 99%) of the vertices of G. Tom Hutchcroft and I recently showed that such a p exists provided G is not close to a cycle in some sense, confirming most cases of a conjecture of Benjamini (2001). I will give an overview of the proof, and go into some detail on certain aspects of it that have a distinctly additive combinatorial flavour.