Given a (monotone) graph property P and a random graph G(n,m) with n vertices and m edges, typically, there is a value m(n) such that P holds (does not hold) with probability approaching 1 when m is sufficiently above (below) m(n). The value m(n) is called a threshold value for P and usually there is a small window around this value where the transition occurs. These sudden changes in the behavior of graphs, known as phase transitions, are widely observed in random graph processes. Investigating thresholds and transition windows for various graph properties is a central theme of study in the field of random graph theory.

I will talk about phase transitions in random chord diagrams and random permutations.

A chord diagram of size n is a pairing of 2n points lying on a circle. By connecting the points in each pair, we obtain n chords. This gives a graph, called the chord intersection graph, where the vertices and edges of the graph correspond to the chords and crossings of the chord diagram, respectively. Hence we can talk about the graph theoretic properties of chord diagrams such as connectedness or the size of a largest component. A permutation may be viewed as a chord diagram where all n chords cross an additional chord that is not part of the diagram. In particular, I will talk about the appearance of a big component in random chord diagrams and a connectedness threshold for random permutations.

Joint work with Boris Pittel.