A number of models arising in statistical physics have the following structure. Each site in a (possibly infinite) graph has to have one of a fixed set of "spins". Certain pairs of spins may not occur on adjacent sites in the graph. One then looks for "nice" measures on the set of all legal configurations: if there is more than one then we speak of a "phase transition".

One of the simplest such models is that of proper colourings of a graph (i.e., colourings of the graph in which adjacent sites get different colours) with q colours. In statistical physics, this is the q-state Potts model at zero temperature. Even the problem of colouring the r-regular infinite tree turns out to be interesting and challenging.

We introduce two probabilistic models for N interacting Brownian motions moving in a trap in R^d under the presence of mutually repellent forces. The two models are defined in terms of transformed path measures on finite time intervals under a trap Hamiltonian and two respective pair-interaction Hamiltonians. The first pair interaction exhibits a particle repellence, while the second one imposes a path repellency.

We analyse both models in the limit of diverging time with fixed number N of Brownian motions. In particular, we prove large deviations principles for the normalised occupation measures. The minimisers of the rate functions in the two cases are given in terms of the ground state, respectively the ground product-states, of a certain associated operator, the Hamilton operator for a system of N interacting trapped particles (bosons). In the case of path-repellency, we also discuss the case of a Dirac-type interaction, which is rigorously defined in terms of Brownian intersection local times. We prove a large-deviation result for a discrete toy model. This study is a contribution to the search for a mathematical formulation of the quantum system of N trapped interacting bosons as a model for Bose-Einstein condensation, motivated by the success of the famous 1995 experiments. Recently, Lieb et al. described the large-N behaviour of the unrestricted ground states in terms of the well-known Gross-Pitaevskii formula, involving the scattering length of the pair potential. We prove that the large-N behaviour of the product-state ground states is also described by the Gross-Pitaevskii formula, however with the scattering length of the pair potential replaced by its integral.

Anderson localisation is an important phenomenon describing a transition between insulation and conductivity. The problem is to analyse the spectrum of a Schroedinger operator with a random potential in the Euclidean space or on a lattice. We say that the system exhibits (exponential) localisation if with probability one the spectrum is pure point and the corresponding eigen-functions decay exponentially fast.

So far in the literature one considered a single-particle model where the potential at different sites is IID or has a controlled decay of correlations. I'll discuss some new results on $N$-particle Anderson model (Wegner's estimates and the multi-scale analysis scheme).

This is a joint work with V Chulaevsky (University of Reims, France)

We consider walkers on a two-dimensional lattice making random decisions about the steps they take, based on information from their immediate surroundings only. It was shown recently that two such random walkers, although taking decisions in very different ways, yet explore the same area of space. Motivated by this result, we describe a specific class of random walkers whose continuum limit is reflected Brownian motion, and investigate some of their intriguing properties.

The symmetry set (SS) of a plane curve, is the closure of the set of centres of circles which are tangent to the curve at two different places, at least. The medial axis (MA) is the subset of the SS consisting of the closure of the locus of centres of circles which are maximal, i.e whose radii are the minimum distance from their centres to the curve.

We will give a little survey, and discuss new results about the evolution of SS/MA of families of plane curves which include singular members, as well as some applications in Computer Vision and Image Analysis.