2008/09 - Term 2
| Amanda Turner (Lancaster University )
Planar aggregation and the coalescing Brownian flow
Diffusion limited aggregation (DLA) is a random growth model which was originally introduced in 1981 by Witten and Sander. This model is prevalent in nature and has many applications in the physical sciences as well as industrial processes. Unfortunately it is notoriously difficult to understand, and only one rigorous result has been proved in the last 25 years. We consider a simplified version of the Hastings-Levitov model of planar aggregation which is obtained by composing certain independent random conformal maps and show that the evolution of harmonic measure on the boundary of the cluster converges to the coalescing Brownian flow.
| no seminar
| Jean-Bernard Bru (University of Vienna)
Equilibrium States of Fermi Systems, Superconductivity
I will first present a rigorous derivation of a generalized Bogoliubov approximation for a large class of Fermi systems. This analysis leads to a description of equilibrium states. As an example, I with then explain the equilibrium and ground states of the strong coupling BCS--Hubbard Hamiltonian. This study highlights the thermodynamic impact of the Coulomb repulsion on s--wave superconductors: the Meissner effect is shown to be rather generic but coexistence of superconducting and ferromagnetic phases is also shown to be feasible. Our proof of a superconductor--Mott insulator phase transition implies a rigorous explanation of the necessity of doping insulators to create superconductors. These mathematical results are consequences of “quantum large deviation arguments” combined with an adaptation of the proof of Störmer theorem to even states on the CAR algebra.
| Benjamin Schlein (Cambridge University)
Derivation of the Gross-Pitaevskii equation
In this talk I am going to present a rigorous derivation of the time-dependent Gross-Pitaevskii equation for the description of the time-evolution of Bose-Einstein condensates starting from first principle many body quantum dynamics.
| Tomohiro Sasamoto (TU Munich)
The fluctuations of the 1D polynuclear growth model
We discuss fluctuation properties of a one dimensional stochastic surface growth process called the polynuclear growth model. When the surface grows into a droplet shape, the fluctuations are the same as those of the largest eigenvalue of GUE from random matrix theory. When the growth is flat in average, the fluctuations change to what we call the Airy_1 process. This is shown using the connection to the totally asymmetric exclusion process.
| Manfred Salmhofer (University of Heidelberg)
Analysis of functional integrals for many-fermion systems
I review the general scope of the study of fermionic many-body systems by functional Berezin integrals and discuss the main techniques for their mathematical control, as well as some results obtained by this method.
| Nicolas Petrelis (TU-Berlin)
Copolymer in an emulsion: supercritical and subcritical regime
In this talk we discuss a two-dimensional directed self-avoiding walk model of a random copolymer in a random emulsion. The copolymer is a random concatenation of monomers of two types, A and B, each occurring with density 1/2. The emulsion is a random mixture of liquids of two types, A and B, organised in large square blocks occurring with density p and 1-p, respectively, where p\in (0,1). The copolymer in the emulsion has an energy that is minus \alpha times the number of AA-matches minus \beta times the number of BB-matches. We will consider both the supercritical regime (oil droplets form an infinite cluster) and the subcritical regime (no infinite cluster).
| Frank Redig (Universiteit Leiden)
A new view on duality in interacting particle systems
We show how duality arises naturally from extra symmetries in the generator of a Markov process. We illustrate this approach in the context of exclusion processes (SU(2) symmetry) as well as in the context of a model of interacting diffusions (SU(1,1) symmetry). We show how the duality of the Kipnis-Marchioro-Presutti model naturally fits in this context, and how the dual functions can be constructed from the underlying symmetry. The talk is based on joint work with C. Giardina, J. Kurchan and K. Vafayi.
| Peter Moerters (University of Bath)
Upper tails for intersection local times of random walks in supercritical dimensions
We determine the precise asymptotics of the upper tail probability of the total intersection local time of p independent random walks on the d-dimensional lattice under the assumption p(d ? 2) > d. Our approach allows a direct treatment of the infinite time horizon
| Hugo Touchette (Queen Mary, University of London)
Nonconcave entropies in statistical mechanics
Many-body systems involving long-range interactions, such as self-gravitating particles or unscreened plasmas, give rise to equilibrium and nonequilibrium properties that are not seen in short-range systems. One such property is that long-range systems can have negative heat capacities, which implies that these systems cool down by absorbing energy. This talk will discuss the origin of this unusual property, as well as some of its connections with phase transitions, metastability, and the nonequivalence of statistical ensembles. It will be seen that the essential difference between long- and short-range systems is that the entropy can be nonconcave as a function of the energy for long-range systems. For short-range systems, the entropy is always concave.