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