Derrida et al. derived (in the 1990s) some asymptotic formulae for the probabilities of large gaps in infinite coalescing and/or annihilating random walks in dimension $d=1$ (and also a persistence exponent). These particle systems are now known to be integrable and these answers can be derived by manipulating determinants of certain integral operators that underly their algebraic structure. Such manipulations were first done by Tracy and Widom for random matrix models.

Exactly the same problems arise when studying certain random polynomials models (Kac polynomials).

The random transposition shuffle is defined by our hands indepedently choosing a card each to transpose every step. The left-right shuffle follows from the modifcation of our hands now being dependent and not able to cross each other. We show how a card shuffle may be viewed as a random walk on the symmetric group and give a description of how algebraic techniques can be used to analyse the speed at which this random walk converges to the uniform distribution. We uncovered a remarkable branching structure involving Young diagrams which allows us to label the eigenvalues for this shuffle. After analysis of the eigenvalues we find the number of shuffles required to get close to uniform is $n\log(n) +cn$.

The inhomogeneous Curie–Weiss model is an interacting (Ising) spin model on the complete graph. The inhomogeneity is introduced by adding a positive weight to every vertex and letting the interaction strength between spins at two vertices be proportional to the product of their weights. We explain how this model arises in the study of Ising models on random graphs and investigate several limit theorems for this model. We especially look at the case where the spins are unbounded and what effects the tail of the weight distribution has on the behaviour of the model.

This talk concerns the long term behaviour of a growth model with graph-based interaction. The model describes a random sequential allocation of particles at vertices of a finite graph and can be regarded as a variant of interacting urn model.