## November 3, 2023

Location: Janskerkhof 15a, room 101.We analyse the mixing profile of a random walk on a dynamic random permutation, focusing on the regime where the walk evolves much faster than the permutation. We consider two types of dynamics: one allows for coagulation of permutation cycles only, the other allows for both coagulation and fragmentation. We show that for both types, after scaling time by the length of the permutation and letting this length tend to infinity, the total variation distance between the current distribution and the uniform distribution converges to a limit process that drops down in a single jump. This jump is similar to a one-sided cut-off, occurs after a random time, and goes from the value 1 to a value that is a strictly decreasing and deterministic function of the time of the jump, related to the size of the largest component in Erdős-Rényi random graphs. After the jump, the total variation distance follows this function down to 0. Joint work with the trio of my doctoral advisors: Luca Avena (Florence), Remco van der Hofstad (TU/e) and Frank den Hollander (Leiden).

Several real-life networks exhibit power law degree distribution and dynamic characteristics. In my presentation, I will delve into the intriguing world of dynamic sparse random graphs, particularly focusing on those with power law degree distributions, such as the preferential attachment model.

In this model, each new vertex joins the graph with an independently and identically distributed (i.i.d.) random number of edges and connects to existing vertices with a probability proportional to the degree of the vertex at that moment. My talk is divided into two key components:

1. Local Limit Analysis: I will explore the local limit of these random graphs. Although there are several models in the literature depending on slight variations in the attachment rules, we prove all of these models share a common locally tree-like structure.

2. Critical Percolation Threshold: Building on the understanding of the local limit, we investigate the critical percolation threshold of these graphs. Using the large set expanders with bounded average degrees property of preferential attachment models, I will discuss how we can apply this property to study the critical percolation threshold on the local limit. I will present our explicit calculation of the critical percolation threshold, using the spine decomposition method and other fascinating analytical tools. You can expect to hear about some intriguing and unexpected results regarding this critical percolation threshold.

Open quantum systems under continuous observation are modelled as "quantum trajectories". These are stochastic processes on state space, consisting of Brownian-type motion and "quantum jumps". A good entry to these processes was found in the 1980's by Davies, Barchielli, and Holevo: they stressed the infinitely divisible character of continuous observation, and found a convolution semigroup of "instruments" for its description. A full classification of these semigroups and their characteristic exponents was provided by Holevo, using dilations of positive definite kernels. For a Festschrift on the occasion of his 80-th birthday Burkhard Kümmerer and myself presented a new proof of the classification theorem, based on weak convergence of completely positive measures. This will also form a chapter in our book on Quantum Markov Processes.