Simulations of uniformly random domino tilings of large Aztec diamonds give striking pictures due to the emergence of two macroscopic regions. These regions are often referred to as solid and liquid phases. A limiting curve separates these regions and interesting probabilistic features occur around this curve, which are related to random matrix theory. The two-periodic Aztec diamond features a third phase, often called the gas phase. In this talk, we introduce the model and discuss some of the asymptotic behavior at the liquid-gas boundary.

This is based on joint works with Vincent Beffara (Grenoble), Kurt Johansson (Stockholm) and Benjamin Young (Oregon).

Although animal locations gained via GPS, etc. are typically observed on a discrete time scale, movement models formulated in continuous time are preferable; avoiding the struggles experienced in discrete time when faced with irregular observations or the prospect of comparing analyses on different time scales. A class of models able to emulate a range of movement ideas are defined by representing movement as a combination of stochastic processes describing both speed and bearing. This framework can then be extended to allow multiple behavioural modes through a continuous time Markov process. Bayesian inference for such models uses a hybrid MCMC approach, relying on augmenting observed locations with a more detailed movement path gained via simulation techniques. This method of inference will be illustrated with both simulated and real data.

We first attach a uniformly distributed random variable x to each vertex which we call a vertex's location. Having selected a set of vertices using preferential attachment the new vertex attaches to a vertex contained in the selected set which has the optimal location according to predetermined criteria. We are exploring when the condensation phenomenon occurs. The two criteria we have considered are (a) the median location of three, and (b) the second or sixth of seven each with probability 0.5. Further work we are currently looking at extending the location to a two dimensional vector oppose to a scalar. This work is based on the work of Jordan [Preprint] where the criteria is defined as the largest location.

Unlike Markov chain Monte Carlo, perfect simulation algorithms produce a sample from the exact equilibrium distribution of a Markov chain, but at the expense of a random run-time. The last few years have seen much progress in the area of perfect simulation algorithms for multi-server queueing systems. I'll give a short introduction to perfect simulation algorithms for beginners, before talking about some recent work on designing a so-called "omnithermal" algorithm for M/G/c queues, which allows us to sample simultaneously from the equilibrium distributions for a range of c (the number of servers).

Geometric random graph models aim to more realistically approximate real-world networks for which the underlying geometry might be expected to play a role. One disadvantage is that, even for a given geometry, there is typically no single natural or canonical model. The group-walk random graph model gives a natural construction for link formation where the underlying geometry is the Cayley graph of a finitely-generated group, with strong connections to the Poisson boundary and long-range percolation. We focus on the case of the free group on two generators. In contrast to some other groups, percolation almost surely does not occur for any intensity of the process. We show that the expected size of a cluster is finite, and give asymptotic bounds. For the finite version of the model we prove a sharp threshold for connectedness; perhaps surprisingly, isolated vertices are not the main obstacle to connectivity.

This is joint work with Agelos Georgakopoulos (Warwick).