MARK KAC SEMINAR

I will first explain how the study of the lattice fields associated with certain classical statistical mechanics models naturally leads to consider random loop models. I will then argue that these loop models are interesting in their own right, and I will show how they can be used to define new random fields, completing the loop. (Partly based on joint work with Marcin Lis.)

Scale-free percolation, as introduced by Deijfen, van der Hofstad, and Hooghiemstra (2013) denotes a percolation model on Z^d, where two points x and y are connected by an edge with probability

1-exp{-p W_x W_y / dist(x,y)^b }

where p>0 is a percolation parameter, W_x and W_y are i.i.d. edge weights with power law distribution, and b denotes the exponent for the long-range connections. The specific features of scale-free percolation are (for suitable parameter ranges):

• the length of a shortest path between two vertices x and y is bounded by O(log|x-y|) (a `small world network’)
• the degrees of vertices exhibit a power law (a `scale-free network’)

After an introduction to the model, I shall focus on the regime where the expected degrees are infinite. I present criteria for transience vs. recurrence, and the emergence of a specific hierarchical network as subgraph of the infinite cluster.

The talk is based on joint work with Joost Jorritsma and Tim Hulshof.