Bernard Souillard

Sur le spectre des opérateurs aux différences finies aléatoires

We study a class of random finite difference operators, a typical example of which is the finite difference Schrödinger operator with a random potential which arises in solid state physics in the tight binding approximation. We obtain with probability one, in various situations, the exact location of the spectrum, and criterions for a given part in the spectrum to be pure point or purely continuous, or for the static electric conductivity to vanish. A general formalism is developped which transforms the study of these random operators into that of the asymptotics of a multiple integral constructed from a given recipe. Finally we apply our criterions and formalism to prove that, with probability one, the one-dimensional finite difference Schrödinger operator with a random potential has pure point spectrum and developps no static conductivity.

LOCALIZATION OF GRAVITY WAVES ON A CHANNEL WITH A RANDOM BOTTOM

We present a theoretical study of the localisation phenomenon of gravity waves by a rough bottom in a one-dimensional channel. After recalling localisation theory and applying it to the shallow-water case, we give the first study of the localisation problem in the framework of the full potential theory; in particular we develop a renormalised-transfer-matrix approach to this problem. Our results also yield numerical estimates of the localisation length, which we compare with the viscous dissipation length. This allows the prediction of which cases localisation should be observable in and in which cases it could be hidden by dissipative mechanisms.

Anderson localization for multi-dimensional systems at large disorder or large energy

We prove that discrete Schrödinger operators on ℤd with a random-potential have almost-surely only pure point spectrum and exponentially decaying eigenfunctions for large disorder or large energy. This is the first proof of localization for multi-dimensional Anderson models.

Essential Singularity in Percolation Problems and Asymptotic Behavior of Cluster Size Distribution

AbstractIt is rigorously proved that the analog of the free energy for the bond and site percolation problem on

Decrease properties of truncated correlation functions and analyticity properties for classical lattices and continuous systems

\mathbb{Z}^v

Strong cluster properties for classical systems with finite range interaction

in arbitrary dimensionΝ (Ν> 1) has a singularity at zero external field as soon as percolation appears, whereas it is analytic for small concentrations. For large concentrations at least, it remains, however, infinitely differentiable and Borel-summable. Results on the asymptotic behavior of the cluster size distribution and its moments, and on the average surface-to-size ratio, are also obtained. Analogous results hold for the cluster generating function of any equilibrium state of a lattice model, including, for example, the Ising model, but infinite-range andn-body interactions are also allowed.

Cluster properties of lattice and continuous systems

We discuss a rotation number α(λ) for second order finite difference operators. Ifk(λ) denotes the integrated density of states, thenk(λ)=2α(λ). For almost periodic operators,k(λ) is proved to lie in the frequency-module whenever λ is outside the spectrum; this yields a labelling of the gaps of the spectrum.

Anderson localization for one- and quasi-one-dimensional systems

We present and discuss some physical hypotheses on the decrease of truncated correlation functions and we show that they imply the analyticity of the thermodynamic limits of the pressure and of all correlation functions with respect to the reciprocal temperature β and the magnetic fieldh (or the chemical potential μ) at all (real) points (β0,h0) (or (β0, μ0)) where they are supposed to hold. A decrease close to our hypotheses is derived in certain particular situations at the end.