## Thomas Vallier (15/09/14)

Math Colloquium

### Speaker: Thomas Vallier, University of Iceland

Title: Bootstrap percolation on the random graph \(G_{n,p}\)

Location: V-157, VRII

Time: Monday September 15, at 15:00-16:00.

### Abstract:

Bootstrap percolation on the random graph \(G_{n,p}\) is a process of spread of “activation” on a given realization of the graph with a given number of initially active nodes. At each step those vertices which have not been active but have at least \(r ≥ 2\) active neighbours become active as well.

We consider the n vertices with global connections inherited from the structure of the graph \(G_{n,p}\), meaning that any two vertices share an edge with probability \(p\) independently of the others.

The presentation is based on the article ”Bootstrap percolation on the random graph \(G_{n,p}\)” by Janson, Luczak, Turova and Vallier.

Among other results, they study the size \(A^*\) of the final active set depending on the number of vertices active at the origin as a function of n (the number of vertices) and \(p = p(n)\) (the probability of connections) which is written \(a_0(n, p) = a_0\). The model exhibits a sharp phase transition: depending on the parameters of the model the final size of activation with a high probability is either \(n − o(n)\) or it is \(o(n)\).

I will give a pictorial introduction to the model and explain briefly the approach of the authors to derive the threshold for bootstrap percolation on \(G_{n,p}\).