Disordered systems and Burgers' turbulence

Talk presented at the International Conference on Mathematical Physics (Brisbane 1997). This is an introduction to recent work on the scaling and intermittency in forced Burgers turbulence. The mapping between Burgers' equation and the problem of a directed polymer in a random medium is used in order to study the fully developped turbulence in the limit of large dimensions. The stirring force corresponds to a quenched (spatio temporal) random potential for the polymer, correlated on large distances. A replica symmetry breaking solution of the polymer problem provides the full probability distribution of the velocity difference $u(r)$ between points separated by a distance $r$ much smaller than the correlation length of the forcing. This exhibits a very strong intermittency which is related to regions of shock waves, in the fluid, and to the existence of metastable states in the directed polymer problem. We also mention some recent computations on the finite dimensional problem, based on various analytical approaches (instantons, operator product expansion, mapping to directed polymers), as well as a conjecture on the relevance of Burgers equation (with the length scale playing the role of time) for the description of the functional renormalisation group flow for the effective pinning potential of a manifold pinned by impurities.Comment: Latex, 11 page

Computation of a universal deformation ring

We compute the universal deformation ring of an odd Galois two dimensional representation of Gal$(M/Q)$ with an upper triangular image, where $M$ is the maximal abelian pro-$p$-extension of $F_{\infty}$ unramified outside a finite set of places S, $F_{\infty}$ being a free pro-$p$-extension of a subextension $F$ of the field $K$ fixed by the kernel of the representation. We establish a link between the latter universal deformation ring and the universal deformation ring of the representation of Gal$(K_S/Q)$, where $K_S$ is the maximal pro-$p$-extension of $K$ unramified outside $S$. We then give some examples. This paper was accepted for publication in the Mathematical Proceedings of the Cambridge philosophical society (May 99)

Elastic Rod Model of a Supercoiled DNA Molecule

We study the elastic behaviour of a supercoiled DNA molecule. The simplest model is that of a rod like chain, involving two elastic constants, the bending and the twist rigidities. We show that this model is singular and needs a small distance cut-off, which is a natural length scale giving the limit of validity of the model, of the order of the double helix pitch. The rod like chain in presence of the cutoff is able to reproduce quantitatively the experimentally observed effects of supercoiling on the elongation-force characteristics, in the small supercoiling regime. An exact solution of the model, using both transfer matrix techniques and its mapping to a quantum mechanics problem, allows to extract, from the experimental data,the value of the twist rigidity. We also analyse the variation of the torque and the writhe to twist ratio versus supercoiling, showing analytically the existence of a rather sharp crossover regime which can be related to the excitation of plectonemic-like structures. Finally we study the extension fluctuations of a stretched and supercoiled DNA molecule, both at fixed torque and at fixed supercoiling angle, and we compare the theoretical predictions to some preliminary experimental data.Comment: 29 pages Revtex 5 figure

Group Testing with Random Pools: optimal two-stage algorithms

We study Probabilistic Group Testing of a set of N items each of which is defective with probability p. We focus on the double limit of small defect probability, p>1, taking either p->0 after $N\to\infty$ or $p=1/N^{\beta}$ with $\beta\in(0,1/2)$. In both settings the optimal number of tests which are required to identify with certainty the defectives via a two-stage procedure, $\bar T(N,p)$, is known to scale as $Np|\log p|$. Here we determine the sharp asymptotic value of $\bar T(N,p)/(Np|\log p|)$ and construct a class of two-stage algorithms over which this optimal value is attained. This is done by choosing a proper bipartite regular graph (of tests and variable nodes) for the first stage of the detection. Furthermore we prove that this optimal value is also attained on average over a random bipartite graph where all variables have the same degree, while the tests have Poisson-distributed degrees. Finally, we improve the existing upper and lower bound for the optimal number of tests in the case $p=1/N^{\beta}$ with $\beta\in[1/2,1)$.Comment: 12 page