I will present an explicit probabilistic construction of Haar measure on the unitary group, and show how that can give a different way to understand characteristic polynomials of random unitary matrices. The motivation for studying characteristic polynomials comes from number theory, as it's believed they model the Riemann zeta function. This work was joint with Paul Bourgade, Ashkan Nikeghbali and Marc Yor.

We consider a stochastic queueing system modelling the behaviour of a wireless network with nodes employing a discrete-time version of the standard decentralised medium access algorithm. The system is unsaturated—each node receives an exogenous flow of packets at the rate $\lambda$ packets per time slot. Each packet takes one slot to transmit, but neighboring nodes cannot transmit simultaneously. The algorithm we study is standard in that a node with empty queue does not compete for medium access and the access procedure by a node does not depend on its queue length, as long as it is non-zero. Two system topologies are considered, with nodes arranged in a circle and in a line. We prove that, for either topology, the system is stochastically stable under condition $\lambda<2/5$. This result is intuitive for the circle topology as the throughput each node receives in a saturated system (with infinite queues) is equal to the so-called parking constant, which is larger than $2/5$. (The latter fact, however, does not help to prove our result.) The result is not intuitive at all for the line topology as in a saturated system some nodes receive a throughput lower than $2/5$. This is joint work with Sasha Stolyar (UIUC).

We will be discussing several load balancing mechanisms based on random allocation protocols and random walks. Our focus will be on making standard models more applicable to load balancing problems, e.g., by allowing to model tasks sizes and processing speeds, or by attempting to "parallelise" inherently sequential-seeming protocols. This will be more of an overview talk light on proofs (though main ideas and techniques will be hinted at).

The many authors involved in the various pieces of work will be duly mentioned during the talk.

Many random processes arising in applications exhibit a range of possible behaviours depending upon the values of certain key factors. Investigating critical behaviour for such systems leads to interesting and challenging mathematics. Much progress has been made over the years using a variety of techniques. This presentation will give a brief introduction to the asymptotic behaviour of the centre of mass of a $d$-dimensional random walk $S_n$, which is defined by $G_n=n^{-1} \sum_{i=1}^{n} S_i$, $n \ge 1$. By considering the local central limit theorem, we investigate the almost-sure asymptotic behaviour of the centre of mass process. We obtain a recurrence result in one dimension under minor moments assumptions; in the case of simple symmetric random walk the fact that $G_n$ returns infinitely often to a neighbourhood of the origin is due to Grill in 1988. We also obtain the transience result for dimensions greater than one. In particular, we give a diffusive rate of escape; again in the case of simple symmetric random walk the result is due to Grill. This is joint work with Andrew Wade (Durham).