On the Geometry of Chemical Reaction Networks: Lyapunov Function and Large Deviations

In an earlier paper, we proved the validity of large deviations theory for the particle approximation of quite general chemical reaction networks. In this paper, we extend its scope and present a more geometric insight into the mechanism of that proof, exploiting the notion of spherical image of the reaction polytope. This allows to view the asymptotic behavior of the vector field describing the mass-action dynamics of chemical reactions as the result of an interaction between the faces of this polytope in different dimensions. We also illustrate some local aspects of the problem in a discussion of Wentzell–Freidlin theory, together with some examples.     

A spin-glass model with random couplings

We define a frustrated spin-glass model for which the Migdal-Kadanoff renormalization group is exact. Our model has random couplings, and the renormalization group acts on these. We study the high and low temperature phases of the model, exhibit a critical fixed point (in high dimension), and show that the Edwards-Anderson parameter takes a non-zero value in the low-temperature phase.On leave from the University of Geneva     

Universal properties of maps on an interval

We consider itcrates of maps of an interval to itself and their stable periodic orbits. When these maps depend on a parameter, one can observe period doubling bifurcations as the parameter is varied. We investigate rigorously those aspects of these bifurcations which are universal, i.e. independent of the choice of a particular one-parameter family. We point out that this universality extends to many other situations such as certain chaotic regimes. We describe the ergodic properties of the maps for which the parameter value equals the limit of the bifurcation points.     

Representations of the CCR in the (φ4)3 model: Independence of space cutoff

The algebra of observables for the renormalized ⁴ interaction in 3-dimensional space-time is constructed. It is shown that the von Neumann algebras associated with observables in a bounded regionB are independent of the space cutoff which is used in the construction of a Hamiltonian for this interaction. This result is shown to be useful in the construction of a translation invariant ⁴ theory in three dimensions. It also gives a physical criterion for the equivalence of non-Fock representations of the canonical commutation relations.     

A Topological Glass

We propose and study a model with glassy behavior. The state space of the model is given by all triangulations of a sphere with n nodes, half of which are red and half are blue. Red nodes want to have 5 neighbors while blue ones want 7. Energies of nodes with other numbers of neighbors are supposed to be positive. The dynamics is that of flipping the diagonal of two adjacent triangles, with a temperature dependent probability. We show that this system has an approach to a steady state which is exponentially slow, and show that the stationary state is unordered. We also study the local energy landscape and show that it has the hierarchical structure known from spin glasses. Finally, we show that the evolution can be described as that of a rarefied gas with spontaneous generation of particles and annihilating collisions.     

A note on the power spectrum of the iterates of Feigenbaum's function

We give rigorous bounds on the scaling behaviour of the power spectrum for Feigenbaum's map.Work partially supported by the Fonds National Suisse     

A global attracting set for the Kuramoto-Sivashinsky equation

New bounds are given for the L²-norm of the solution of the Kuramoto-Sivashinsky equation $$\partial _t U(x,t) = - (\partial _x^2 + \partial _x^4 )U(x,t) - U(x,t)\partial _x U(x,t)$$ , for initial data which are periodic with periodL. There is no requirement on the antisymmetry of the initial data. The result is $$\mathop {\lim \sup }\limits_{t \to \infty } \left\| {U( \cdot ,t)} \right\|_2 \leqslant const. L^{8/5} $$ .     

The largest Liapunov exponent for random matrices and directed polymers in a random environment

We study the largest Liapunov exponent for products of random matrices. The two classes of matrices considered are discrete,d-dimensional Laplacians, with random entries, and symplectic matrices that arise in the study ofd-dimensional lattices of coupled, nonlinear oscillators. We derive bounds on this exponent for all dimensions,d, and we show that ifd3, and the randomness is not too strong, one can obtain an explicit formula for the largest exponent in the thermodynamic limit. Our method is based on an equivalence between this problem and the problem of directed polymers in a random environment.