I will discuss the influence of the seed in models of randomly growing trees; in particular, I will focus on the preferential attachment and uniform attachment models. In both of these models, different seeds lead to different distributions of limiting trees from a total variation point of view. I will discuss the differences and similarities in proving this for the two models. This is based on joint work with Sebastien Bubeck, Ronen Eldan, and Elchanan Mossel.

The probabilistic method is an efficient way to prove the existence of certain combinatorial structures when constructions are not available (or harder to get). For this we define a discrete probability space on the structures and prove that the desired property holds with positive probability. For a long while the method mostly worked if the probability was not only positive but overwhelming (close to 1). While effectively and deterministically find such a structure may still be hard, a randomized algorithm is trivial: just pick a random sample from the space and check if the desired property holds. The Lovász local lemma (1975) gave a tool to prove the probability of a property is positive in many cases even if the probability itself was exponentially small. In these cases finding the desired structure (a needle in the haystack) is challenging even for a randomized algorithm. We show that a very simple and intuitive randomized algorithm does exactly that: it quickly finds a desired structure.

Most of the talk is based on a 2010 joint paper with Robin Moser with pointers for more recent developments if time permits.

The KPZ scaling theory provides a general method to compute all model-dependent constants arising in limit theorems for a large class of exclusion processes. The validity of this heuristic approach is rigorously proved only for a few exactly solvable models. In this talk, we will discuss how the theory applies for q-deformed exclusion processes introduced by Borodin-Corwin and Povolotsky: The q-TASEP and the q-Hahn TASEP. We will also introduce a two-sided generalization of the q-Hahn TASEP that preserve the integrable structure and further confirm KPZ scaling theory. This is a joint work with Ivan Corwin.

We investigate the behaviour of random walk on the circle packing of a hyperbolic unimodular triangulations. A way to relate the geometry of such graphs with random walks is via circle packing: one can draw non-intersecting circles on the plane, one for each vertex and circles touch each other if and only if their vertices are adjacent. We show that such a triangulation can be packed in the unit disc and the unit circle gives a pretty accurate description of the final behaviour of the simple random walk on it. Joint work with Omer Angel, Tom Hutchcroft and Asaf Nachmias.

In the beginning of the talk I will give a short introduction to the frequentist analysis of Bayesian approaches, focusing mainly on nonparametric and adaptive settings. Then I will present our results on the frequentist coverage of Bayesian credible sets in a nonparametric setting. In our work we consider a scale of priors of varying regularity and choose the regularity by an empirical Bayes method. Next we consider a central set of prescribed posterior probability in the posterior distribution of the chosen regularity. We show that such an adaptive Bayes credible set gives correct uncertainty quantification of `polished tail' parameters, in the sense of high probability of coverage of such parameters. On the negative side we also show by theory and example that adaptation of the prior necessarily leads to gross and haphazard uncertainty quantification for some true parameters that are still within the hyperrectangle regularity scale.

We consider a random geometric graph model relevant to wireless mesh networks. Nodes are placed uniformly in a domain, and pairwise connections are made independently with probability a specified function of the distance between the pair of nodes, and in a more general anisotropic model, their orientations. The probability that the network is (k-)connected is estimated as a function of density using a cluster expansion approach. This leads to an understanding of the crucial roles of local boundary effects and of the tail of the pairwise connection function, in contrast to lower density percolation phenomena.

We consider a decentralized traffic signal control policy for urban road networks. The policy uses a proportionally fair scheme to allocate green times at each junction using only local information of queue lengths. At the same time it maintains a cyclic ordering of service phases which is a desirable property for traffic safety. Our model also includes elements of real life traffic behaviour such as the use of switching times and the time dependency of service rates due to the initial setup at the beginning of service phases. We discuss the effect of these phenomena on the set of feasible traffic demands. We also investigate the impact of the chosen cycle lengths on the ensuing waiting times of the vehicles. As our main result, we show that the proportionally fair policy stabilizes the network for any feasible demand. In order to accomplish this, we construct the fluid limit of the underlying stochastic system and give a formal proof of convergence as a generalization of prior results in the context of telecommunication applications. Existing stability results in the literature for proportional fair scheduling cannot be directly applied to our model because of several characteristics that are specific to the application to road traffic networks, including the earlier mentioned cyclic service patterns and the time dependency of service rates.

I will survey some results about the speed of equidistribution of random walks on compact Lie groups and on the group of Euclidean isometries. I will also give an application to smoothness of selfsimilar measures. A random walk is a sequence of random elements of the group obtained by computing the product of several independent random elements.