We study one-dimensional diffusions in a stationary and ergodic random environment. We introduce the environment process, an auxiliary process that describes the environment as it is seen by the random walker. A key issue is the existence of an invariant measure for this process that is absolutely continuous w.r.t. the "static" distribution of the environment, as it provides for instance a law of large numbers. We characterize the existence of such an invariant measure. It exists if and only if either the speed in the law of large numbers does not vanish, or b/a is a.s. the gradient of a stationary function, where $a$ and $b$ are the covariance coefficient resp. the local drift attaches to the diffusion.
We consider $N$ Bosons in a box with volume $N/\rho$ under the presence of a mutually repellent pair potential. Denote by $H_N$ the corresponding Hamilton operator with either zero or periodic boundary condition. The symmetrised trace of $e^{-\beta H_N}$ describes the Bosons at positive temperature $1/\beta$. Our main result is a variational formula for the limiting free energy, for any fixed values of the particle density $\rho$ and the inverse temperature $\beta$, in any dimension. The main tools are a description in terms of a marked Poisson point process and a large-deviation analysis of the stationary empirical field. The resulting variational formula in particular describes the asymptotic cycle structure that is induced by the symmetrisation in the Feynman-Kac formula. We close with a short discussion of the relation to Bose-Einstein condensation. (joint work in progress with S. Adams and A. Collevecchio)
Two-dimensional electron gases subjected to a perpendicular magnetic field display a curious physical effect known as the quantum Hall effect. In order to explain this effect, Laughlin proposed a simple wave function as an approximate ground state. Laughlin's function involves the power of a Vandermonde determinant and a Gaussian weight. This talk presents exact results on the L^2 norm and correlation functions in the limit of infinitely many particles when Laughlin's function is adapted to a cylindrical geometry. The results are also of interest for the classical statistical mechanics of Coulomb gases. The talk is based on joint work with Ruedi Seiler and Elliott H. Lieb.
The anomalous dynamical properties of out-of-equilibrium glassy systems, like an extremely slow relaxation to equilibrium, long-time memory effects or aging, have been studied by physicists since 1970's. However the rigorous understanding of these properties is still rather unsatisfactory. In my talk I will summarise the recent advances in this field: First, some older results obtained for the so-called trap-model dynamics in the Random Energy model, and then a new result on the dynamics of p-spin spin glasses.
Half a decade ago, Praehofer and Spohn discovered the Airy_2 process corner growth model. It appeared to be one of the universal processes, appearing in different models including random matrix theory. More recently, we discovered the analogue for growth on a flat substrate: the Airy_1 process. This process does not describe only the large time surface statistics at a fixed time, but its universality extends to any "space-like paths"! We will present the result using the totally asymmetric simple exclusion process (TASEP) as reference model, for which the extreme special cases of space-like paths are (a) fixed time, and (b) tagged particle.
This talk surveys recent work which develops aspects of classical random matrix theory in the broader framework of matrix ensembles associated to classical symmetric spaces. It is a classical result of Wigner that for an hermitian matrix with independent entries on and above the diagonal, the mean empirical eigenvalue distribution converges weakly to the semicircle law as matrix size tends to infinity. In joint work with Katrin Hofmann-Credner (Bochum), this has been generalized to random matrices taken from all infinitesimal versions of classical symmetric spaces. Like Wigner's, this result is universal in that it only depends on certain assumptions about the moments of the matrix entries, but not on the specifics of their distributions. Joint work with Benoît Collins (Lyon/Ottawa) points into a different direction. Here random vectors of the form $(Tr(A^(1) V),..., Tr(A^(r) V))$ are studied, where V is a uniformly distributed element of a matrix version of a classical compact symmetric space, and the $A^(\nu)$ are deterministic parameter matrices. It is proven that for increasing matrix sizes these random vectors converge to a joint Gaussian limit. This generalizes work of Diaconis et al. on the compact classical groups.
Since the seminal work of Dobrushin, it is now a well known fact that the interface between the two phases of 3D Ising model becomes rigid at low temperatures, in the sense that it deviates only locally from a perfect hyperplane. It appears that some models with infinitely degenerated ground states exhibit these rigid interfaces as well. This new result is a first step towards the conjectured existence of non translation-invariant Gibbs measures for some particular models with continuous symmetry. I will present a general method based on correlation inequalities to get such results.
We present results about energetic and dynamic properties of a spinless quantum particle on the Euclidean plane sujected to a perpendicular random magnetic field of Gaussian type with non-zero mean. The results refer only to the limiting case in which the correlation length is finite along one direction and infinite along the perpendicular direction in the plane.
For supercritical multitype Markov branching processes in continuous time, I consider the type evolution along the lineage leading to a randomly chosen individual living at time $t$.
The main results are almost-sure convergence theorems for both time and population averages of ancestral types (conditioned on non- extinction), and the identification of the mutation process describing the type evolution along such lineages. Important tools are a representation of the family tree in terms of a suitable size-biased tree with trunk, and large deviation theory.
During the past decade, the analysis of correlations in (mathematical) classical statistical mechanics has drawn much on the spectral theory of a deformed Hodge Laplacian, which was introduced by Witten in 1983 in a differential geometrical context. The starting point of this analysis is a resolvent identity which is now often refered to as the "Helffer-Sjoestrand formula", bearing the name of the two analysts who established it about 1994-6. The lecture reviews this past and also current developments.
We study the ac-conductivity in linear response theory in the general framework of ergodic magnetic Schr\"odinger operators. For the Anderson model, if the Fermi energy lies in the localization regime, we prove that the ac-conductivity is bounded by $ C \nu2 (\log \frac 1 \nu)^{d+2}$ at small frequencies $\nu$. This is to be compared to Mott's formula, which predicts the leading term to be $ C \nu2 (\log \frac 1 \nu)^{d+1}$. (This is joint work with Abel Klein and Olivier Lenoble.)
