70 research outputs found

### Branching Interfaces with Infinitely Strong Couplings

A hierarchical froth model of the interface of a random $q$-state Potts
ferromagnet in $2D$ is studied by recursive methods. A fraction $p$ of the
nearest neighbour bonds is made inaccessible to domain walls by infinitely
strong ferromagnetic couplings. Energetic and geometric scaling properties of
the interface are controlled by zero temperature fixed distributions. For
$p<p_c$, the directed percolation threshold, the interface behaves as for
$p=0$, and scaling supports random Ising ($q=2$) critical behavior for all
$q$'s. At $p=p_c$ three regimes are obtained for different ratios of ferro vs.
antiferromagnetic couplings. With rates above a threshold value the interface
is linear ( fractal dimension $d_f=1$) and its energy fluctuations, $\Delta E$
scale with length as $\Delta E\propto L^{\omega}$, with $\omega\simeq 0.48$.
When the threshold is reached the interface branches at all scales and is
fractal ($d_f\simeq 1.046$) with $\omega_c \simeq 0.51$. Thus, at $p_c$,
dilution modifies both low temperature interfacial properties and critical
scaling. Below threshold the interface becomes a probe of the backbone geometry
(\df\simeq{\bar d}\simeq 1.305; $\bar d$ = backbone fractal dimension ),
which even controls energy fluctuations ($\omega\simeq d_f\simeq\bar d$).
Numerical determinations of directed percolation exponents on diamond
hierarchical lattice are also presented.Comment: 16 pages, 3 Postscript figure

### Scaling and efficiency determine the irreversible evolution of a market

In setting up a stochastic description of the time evolution of a financial
index, the challenge consists in devising a model compatible with all stylized
facts emerging from the analysis of financial time series and providing a
reliable basis for simulating such series. Based on constraints imposed by
market efficiency and on an inhomogeneous-time generalization of standard
simple scaling, we propose an analytical model which accounts simultaneously
for empirical results like the linear decorrelation of successive returns, the
power law dependence on time of the volatility autocorrelation function, and
the multiscaling associated to this dependence. In addition, our approach gives
a justification and a quantitative assessment of the irreversible character of
the index dynamics. This irreversibility enters as a key ingredient in a novel
simulation strategy of index evolution which demonstrates the predictive
potential of the model.Comment: 5 pages, 4 figure

### The entropic cost to tie a knot

We estimate by Monte Carlo simulations the configurational entropy of
$N$-steps polygons in the cubic lattice with fixed knot type. By collecting a
rich statistics of configurations with very large values of $N$ we are able to
analyse the asymptotic behaviour of the partition function of the problem for
different knot types. Our results confirm that, in the large $N$ limit, each
prime knot is localized in a small region of the polygon, regardless of the
possible presence of other knots. Each prime knot component may slide along the
unknotted region contributing to the overall configurational entropy with a
term proportional to $\ln N$. Furthermore, we discover that the mere existence
of a knot requires a well defined entropic cost that scales exponentially with
its minimal length. In the case of polygons with composite knots it turns out
that the partition function can be simply factorized in terms that depend only
on prime components with an additional combinatorial factor that takes into
account the statistical property that by interchanging two identical prime knot
components in the polygon the corresponding set of overall configuration
remains unaltered. Finally, the above results allow to conjecture a sequence of
inequalities for the connective constants of polygons whose topology varies
within a given family of composite knot types

### Finite-size scaling in unbiased translocation dynamics

Finite-size scaling arguments naturally lead us to introduce a
coordinate-dependent diffusion coefficient in a Fokker-Planck description of
the late stage dynamics of unbiased polymer translocation through a membrane
pore. The solution for the probability density function of the chemical
coordinate matches the initial-stage subdiffusive regime and takes into account
the equilibrium entropic drive. Precise scaling relations connect the
subdiffusion exponent to the divergence with the polymer length of the
translocation time, and also to the singularity of the probability density
function at the absorbing boundaries. Quantitative comparisons with numerical
simulation data in $d=2$ strongly support the validity of the model and of the
predicted scalings.Comment: Text revision. Supplemental Material adde

### Zipping and collapse of diblock copolymers

Using exact enumeration methods and Monte Carlo simulations we study the
phase diagram relative to the conformational transitions of a two dimensional
diblock copolymer. The polymer is made of two homogeneous strands of monomers
of different species which are joined to each other at one end. We find that
depending on the values of the energy parameters in the model, there is either
a first order collapse from a swollen to a compact phase of spiral type, or a
continuous transition to an intermediate zipped phase followed by a first order
collapse at lower temperatures. Critical exponents of the zipping transition
are computed and their exact values are conjectured on the basis of a mapping
onto percolation geometry, thanks to recent results on path-crossing
probabilities.Comment: 12 pages, RevTeX and 14 PostScript figures include

### Scaling symmetry, renormalization, and time series modeling

We present and discuss a stochastic model of financial assets dynamics based
on the idea of an inverse renormalization group strategy. With this strategy we
construct the multivariate distributions of elementary returns based on the
scaling with time of the probability density of their aggregates. In its
simplest version the model is the product of an endogenous auto-regressive
component and a random rescaling factor designed to embody also exogenous
influences. Mathematical properties like increments' stationarity and
ergodicity can be proven. Thanks to the relatively low number of parameters,
model calibration can be conveniently based on a method of moments, as
exemplified in the case of historical data of the S&P500 index. The calibrated
model accounts very well for many stylized facts, like volatility clustering,
power law decay of the volatility autocorrelation function, and multiscaling
with time of the aggregated return distribution. In agreement with empirical
evidence in finance, the dynamics is not invariant under time reversal and,
with suitable generalizations, skewness of the return distribution and leverage
effects can be included. The analytical tractability of the model opens
interesting perspectives for applications, for instance in terms of obtaining
closed formulas for derivative pricing. Further important features are: The
possibility of making contact, in certain limits, with auto-regressive models
widely used in finance; The possibility of partially resolving the long-memory
and short-memory components of the volatility, with consistent results when
applied to historical series.Comment: Main text (17 pages, 13 figures) plus Supplementary Material (16
pages, 5 figures

### A simple model of DNA denaturation and mutually avoiding walks statistics

Recently Garel, Monthus and Orland (Europhys. Lett. v 55, 132 (2001))
considered a model of DNA denaturation in which excluded volume effects within
each strand are neglected, while mutual avoidance is included. Using an
approximate scheme they found a first order denaturation. We show that a first
order transition for this model follows from exact results for the statistics
of two mutually avoiding random walks, whose reunion exponent is c > 2, both in
two and three dimensions. Analytical estimates of c due to the interactions
with other denaturated loops, as well as numerical calculations, indicate that
the transition is even sharper than in models where excluded volume effects are
fully incorporated. The probability distribution of distances between
homologous base pairs decays as a power law at the transition.Comment: 7 Pages, RevTeX, 8 Figure

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