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

The secondary structure of RNA under tension

We study the force-induced unfolding of random disordered RNA or single-stranded DNA polymers. The system undergoes a second order phase transition from a collapsed globular phase at low forces to an extensive necklace phase with a macroscopic end-to-end distance at high forces. At low temperatures, the sequence inhomogeneities modify the critical behaviour. We provide numerical evidence for the universality of the critical exponents which, by extrapolation of the scaling laws to zero force, contain useful information on the ground state (f=0) properties. This provides a good method for quantitative studies of scaling exponents characterizing the collapsed globule. In order to get rid of the blurring effect of thermal fluctuations we restrict ourselves to the groundstate at fixed external force. We analyze the statistics of rearrangements, in particular below the critical force, and point out its implications for force-extension experiments on single molecules.Comment: to be published in Europhys. J.

Nonlinear screening theory of the Coulomb glass

A nonlinear screening theory is formulated to study the problem of gap formation and its relation to glassy freezing in classical Coulomb glasses. We find that a pseudo-gap ("plasma dip") in a single-particle density of states begins to open already at temperatures comparable to the Coulomb energy. This phenomenon is shown to reflect the emergence of short range correlations in a liquid (plasma) phase, a process which occurs even in the absence of disorder. Glassy ordering emerges when disorder is present, but this occurs only at temperatures more then an order of magnitude lower, which is shown to follow from nonlinear screening of the Coulomb interaction. Our result suggest that the formation of the "plasma dip" at high temperatures is a process distinct from the formation of the Efros-Shklovskii (ES) pseudo-gap, which in our model emerges only within the glassy phase.Comment: 5 pages, 2 figures, accepted for publication to Phys. Rev. Let

Message passing algorithms for non-linear nodes and data compression

The use of parity-check gates in information theory has proved to be very efficient. In particular, error correcting codes based on parity checks over low-density graphs show excellent performances. Another basic issue of information theory, namely data compression, can be addressed in a similar way by a kind of dual approach. The theoretical performance of such a Parity Source Coder can attain the optimal limit predicted by the general rate-distortion theory. However, in order to turn this approach into an efficient compression code (with fast encoding/decoding algorithms) one must depart from parity checks and use some general random gates. By taking advantage of analytical approaches from the statistical physics of disordered systems and SP-like message passing algorithms, we construct a compressor based on low-density non-linear gates with a very good theoretical and practical performance.Comment: 13 pages, European Conference on Complex Systems, Paris (Nov 2005

Cavity approach to the spectral density of non-Hermitian sparse matrices

The spectral densities of ensembles of non-Hermitian sparse random matrices are analysed using the cavity method. We present a set of equations from which the spectral density of a given ensemble can be efficiently and exactly calculated. Within this approach, the generalised Girko's law is recovered easily. We compare our results with direct diagonalisation for a number of random matrix ensembles, finding excellent agreement.Comment: 4 pages, 3 figure

SCALING AND INTERMITTENCY IN BURGERS' TURBULENCE

We use the mapping between Burgers' equation and the problem of a directed polymer in a random medium in order to study the fully developped turbulence in the $N$ dimensional forced Burgers' equation. The stirring force corresponds to a quenched (spatio temporal) random potential for the polymer. The properties of the inertial regime are deduced from a study of the directed polymer on length scales smaller than the correlation length of the potential. From this study we propose an Ansatz for the velocity field in the large Reynolds number limit of the forced Burgers' equation in $N$ dimensions. This Ansatz allows us to compute exactly 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. We find that the moments $$ scale as $r^{\zeta(q)}$ with $\zeta(q) \equiv 1$ for all $q \geq 1$. This strong `intermittency' is related to the large scale singularities of the velocity field, which is concentrated on a $N-1$ dimensional froth-like structure.Comment: 35 pages latex, 4 ps figures in separate uufiles package

Spanning Trees in Random Satisfiability Problems

Working with tree graphs is always easier than with loopy ones and spanning trees are the closest tree-like structures to a given graph. We find a correspondence between the solutions of random K-satisfiability problem and those of spanning trees in the associated factor graph. We introduce a modified survey propagation algorithm which returns null edges of the factor graph and helps us to find satisfiable spanning trees. This allows us to study organization of satisfiable spanning trees in the space spanned by spanning trees.Comment: 12 pages, 5 figures, published versio