24 October 2012, Royal Statistical Society, 12 Errol Street, London, EC1Y 8LX

Speakers:

• Bálint Tóth (Technical University Budapest and University of Bristol)
  
  Scaling limits for self-repelling random walks and diffusions with long memory, slides
  
  I will survey recent results about diffusive and superdiffusive asymptotic behaviour of self-interacting random motions, where the pathwise self-interaction is defined in terms of the occupation time measure and it is self-repelling. The asymptotic behaviour is dimension dependent: superdiffusive in one and two dimensions, diffusive in three and more.


• James Martin (University of Oxford)
  
  Particle systems, queues, and models of random growth
  
  I'll survey a collection of inter-related models from probability theory, including interacting particle systems such as exclusion processes, and models of random growth such as first-passage percolation. Multi-type versions of these models have been widely studied recently, with motivations coming from various directions including combinatorics, biology and physics; paths of second-class particles in an exclusion process correspond to competition interfaces between competing populations in spatial growth models. A variety of Markovian queueing systems, which are very natural from an applied probability perspective, turn out to be important in the analysis of these multi-type systems.


• Stefan Grosskinsky (University of Warwick)
  
  Condensation and metastability in stochastic particle systems, slides
  
  The inclusion process is an interacting particle system where particles on connected sites attract each other in addition to performing independent random walks. The system has stationary product measures and exhibits condensation in the limit of strong interactions, where all particles concentrate on a single lattice site. We study the equilibration dynamics on finite lattices in the limit of infinitely many particles, which, in addition to jumps of whole clusters, contains an interesting continuous mass exchange between clusters given by Wright-Fisher diffusions. During equilibration the number of clusters decreases monotonically, and the stationary dynamics consist of jumps of a single remaining cluster (the condensate). This is joint work with Frank Redig and Kiamars Vafayi.


• James Gleeson (University of Limerick)
  
  Approximation methods for binary-state dynamics on complex networks, slides
  
  A wide class of binary-state dynamics on networks---including, for example, the voter model, the Bass diffusion model, and threshold models---can be described in terms of transition rates (spin-flip probabilities) that depend on the number of nearest neighbours in each of the two possible states. High-accuracy approximations for the emergent dynamics of such models on uncorrelated, infinite networks are given by recently-developed compartmental models or approximate master equations (AME). Pair approximations (PA) and mean-field theories can be systematically derived from the AME; we show that PA and AME solutions can coincide in certain circumstances. This facilitates bifurcation analysis, yielding explicit expressions for the critical (ferromagnetic/paramagnetic transition) point of such dynamics, closely analogous to the critical temperature of the Ising spin model.