Diameter in ultra-small scale-free random graphs.

It is well known that many random graphs with infinite variance degrees are ultra-small. More precisely, for configuration models and preferential attachment models where the proportion of vertices of degree at least k is approximately k with τ ∈ (2,3), typical distances between pairs of vertices in a graph of size n are asymptotic to and , respectively. In this paper, we investigate the behavior of the diameter in such models.

The continuum disordered pinning model.

Any renewal processes on [Formula: see text] with a polynomial tail, with exponent [Formula: see text], has a non-trivial scaling limit, known as the [Formula: see text]-. In this paper we consider Gibbs transformations of such renewal processes in an i.i.