François Dunlop

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.

Partial to Complete Wetting: A Microscopic Derivation of the Young Relation

This paper is devoted to the study of the Young equation, which gives a connection between surface tensions and contact angle. We derive the generalized form of this equation for anisotropic models using thermodynamic considerations. In two dimensions with SOS-like approximations of the interface, we prove that the surface tension may be computed explicitly as a simple integral, which of course depends upon the orientation of the interface. This allows a complete study of the wetting transition when a constant wall “attraction” is taken into account within the SOS and Gaussian models. We therefore give a complete analysis of the variation of the contact angle with the temperature for those models. It is found that for certain values of the parameters, two wetting transitions may successively appear, one at low temperature and one at high temperature, giving the following states: film—droplet—film. This study rests upon the generalized Young equation, the validity of which is proved for the Gaussian model with a constant wall attraction, using microscopic considerations.

Pinning of an interface by a weak potential

We prove that in a two-dimensional Gaussian SOS model with a small attractive potential the height of the interface remains bounded no matter how small the potential is; this is in sharp contrast with the free situation in which the interface height diverges logarithmically in the thermodynamic limit.

Zeros of the Partition Function and Gaussian Inequalities for the Plane Rotator Model

The partition function for ferromagnetic plane rotators in a complex external field μ, with ¦Im μ¦ ⩽ ¦Re μ ¦, is bounded below in modulus by its value at μ=0. The proof is based on complex combinations of duplicated variables which are positive definite on a subgroup of the configuration group. In the isotropic situation (and μ=0), the associated “Gaussian inequalities” imply that all truncated correlation functions decay at least as the two-point function.

n-Particle-irreducible functions in euclidean quantum field theory

Abstract We propose equations defining n -particle-irreducible functions for n 1 incoming fields and n 2 outgoing fields, with n ⩾ n 2 ⩾ n 1 . For n = n 2 , or for n ⩽ 5, in a finite volume lattice approximation to a P ( Φ ) theory, we prove that the appropriate “Spencer derivatives” vanish, which implies n -particle decay for weak coupling in two dimensions. These results, extending the results of Spencer for n = 2, are meant to prepare the ground for studying asymptotic completeness in the n -body region, with n arbitrary.

Localization of surface waves on a rough bottom: theories and experiments

We present the first evidences for the localization of water waves on a rough bottom. This is achieved through a very precise experimental setting, as well as an extension of localization theory to the full potential theory of hydrodynamics. For the first time the resonant modes due to the disorder are directly observed experimentally. Preliminary nonlinear results are presented.

Random Walk Weakly Attracted to a Wall

AbstractWe consider a random walk Xn in ℤ+, starting at X0=x≥0, with transition probabilities

P(\Phi)_{2}

\mathbb{P}(X_{n+1}=X_{n}\pm1|X_{n}=y\ge1)={1\over2}\mp{\delta\over4y+2\delta}