Giampaolo Cristadoro

On the origin of long-range correlations in texts

The complexity of human interactions with social and natural phenomena is mirrored in the way we describe our experiences through natural language. In order to retain and convey such a high dimensional information, the statistical properties of our linguistic output has to be highly correlated in time. An example are the robust observations, still largely not understood, of correlations on arbitrary long scales in literary texts. In this paper we explain how long-range correlations flow from highly structured linguistic levels down to the building blocks of a text (words, letters, etc..). By combining calculations and data analysis we show that correlations take form of a bursty sequence of events once we approach the semantically relevant topics of the text. The mechanisms we identify are fairly general and can be equally applied to other hierarchical settings.

Statistical properties of intermittent maps with unbounded derivative

We study the ergodic and statistical properties of a class of maps of the circle and of the interval of Lorenz type which present indifferent fixed points and points with unbounded derivative. These maps have been previously investigated in the physics literature. We prove in particular that correlations decay polynomially, and that suitable limit theorems (convergence to stable laws or central limit theorem) hold for Holder continuous observables. Moreover, we show that the return and hitting times are exponentially distributed in the limit.

Recurrence for quenched random Lorentz tubes

We consider the billiard dynamics in a striplike set that is tessellated by countably many translated copies of the same polygon. A random configuration of semidispersing scatterers is placed in each copy. The ensemble of dynamical systems thus defined, one for each global choice of scatterers, is called quenched random Lorentz tube. We prove that under general conditions, almost every system in the ensemble is recurrent.

Periodic orbit theory of strongly anomalous transport

We establish a deterministic technique to investigate transport moments of an arbitrary order. The theory is applied to the analysis of different kinds of intermittent one-dimensional maps and the Lorentz gas with infinite horizon: the typical appearance of phase transitions in the spectrum of transport exponents is explained.

Measuring logarithmic corrections to normal diffusion in infinite-horizon billiards.

We perform numerical measurements of the moments of the position of a tracer particle in a two-dimensional periodic billiard model (Lorentz gas) with infinite corridors. This model is known to exhibit a weak form of superdiffusion, in the sense that there is a logarithmic correction to the linear growth in time of the mean-squared displacement. We show numerically that this expected asymptotic behavior is easily overwhelmed by the subleading linear growth throughout the time range accessible to numerical simulations. We compare our simulations to analytical results for the variance of the anomalously rescaled limiting normal distributions.

Fractal diffusion coefficient from dynamical zeta functions

Dynamical zeta functions provide a powerful method to analyse low-dimensional dynamical systems when the underlying symbolic dynamics is under control. On the other hand, even simple one-dimensional maps can show an intricate structure of the grammar rules that may lead to a non-smooth dependence of global observables on parameters changes. A paradigmatic example is the fractal diffusion coefficient arising in a simple piecewise linear one-dimensional map of the real line. Using the Baladi–Ruelle generalization of the Milnor–Thurnston kneading determinant, we provide the exact dynamical zeta function for such a map and compute the diffusion coefficient from its smallest zero.

Nonequilibrium stationary states with ratchet effect

An ensemble of particles in thermal equilibrium at temperature T, modeled by Nosè-Hoover dynamics, moves on a triangular lattice of oriented semidisk elastic scatterers. Despite the scatterer asymmetry, a directed transport is clearly ruled out by the second law of thermodynamics. Introduction of a polarized zero mean monochromatic field creates a directed stationary flow with nontrivial dependence on temperature and field parameters. We give a theoretical estimate of directed current induced by a microwave field in an antidot superlattice in semiconductor heterostructures.

Random Walks in a One-Dimensional Lévy Random Environment

We consider a generalization of a one-dimensional stochastic process known in the physical literature as Lévy-Lorentz gas. The process describes the motion of a particle on the real line in the presence of a random array of marked points, whose nearest-neighbor distances are i.i.d. and long-tailed (with finite mean but possibly infinite variance). The motion is a continuous-time, constant-speed interpolation of a symmetric random walk on the marked points. We first study the quenched random walk on the point process, proving the CLT and the convergence of all the accordingly rescaled moments. Then we derive the quenched and annealed CLTs for the continuous-time process.

Follow the fugitive: an application of the method of images to open systems

Borrowing and extending the method of images we introduce a theoretical framework that greatly simplifies analytical and numerical investigations of the escape rate in open systems. As an example, we explicitly derive the exact size- and position-dependent escape rate in a Markov case for holes of finite-size. Moreover, a general relation between the transfer operators of the closed and corresponding open systems, together with the generating function of the probability of return to the hole is derived. This relation is then used to compute the small hole asymptotic behavior, in terms of readily calculable quantities. As an example we derive logarithmic corrections in the second order term. Being valid for Markov systems, our framework can find application in many areas of the physical sciences such as information theory, network theory, quantum Weyl law and, via Ulam?s method, can be used as an approximation method in general dynamical systems.

Characterization of DNA methylation as a function of biological complexity via dinucleotide inter-distances

We perform a statistical study of the distances between successive occurrences of a given dinucleotide in the DNA sequence for a number of organisms of different complexity. Our analysis highlights peculiar features of the CG dinucleotide distribution in mammalian DNA, pointing towards a connection with the role of such dinucleotide in DNA methylation. While the CG distributions of mammals exhibit exponential tails with comparable parameters, the picture for the other organisms studied (e.g. fish, insects, bacteria and viruses) is more heterogeneous, possibly because in these organisms DNA methylation has different functional roles. Our analysis suggests that the distribution of the distances between CG dinucleotides provides useful insights into characterizing and classifying organisms in terms of methylation functionalities.