845 research outputs found

### 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

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