Hamiltonian systems with a mixed phase space typically exhibit an algebraic decay of correlations and of Poincaré recurrences, with numerical experiments over finite times showing system-dependent power-law exponents. We conjecture the existence of a universal asymptotic decay based on results for a Markov tree model with random scaling factors for the transition probabilities. Numerical simulations for different Hamiltonian systems support this conjecture and permit the determination of the universal exponent.

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.complex systems | language dynamics | long correlations | statistical physics | burstiness L iterary texts are an expression of the natural language ability to project complex and high-dimensional phenomena into a one-dimensional, semantically meaningful sequence of symbols. For this projection to be successful, such sequences have to encode the information in form of structured patterns, such as correlations on arbitrarily long scales (1, 2). Understanding how language processes long-range correlations, an ubiquitous signature of complexity present in human activities (3-7) and in the natural world (8-11), is an important task towards comprehending how natural language works and evolves. This understanding is also crucial to improve the increasingly important applications of information theory and statistical natural language processing, which are mostly based on short-range-correlations methods (12-15).Take your favorite novel and consider the binary sequence obtained by mapping each vowel into a 1 and all other symbols into a 0. One can easily detect structures on neighboring bits, and we certainly expect some repetition patterns on the size of words. But one should certainly be surprised and intrigued when discovering that there are structures (or memory) after several pages or even on arbitrary large scales of this binary sequence. In the last twenty years, similar observations of long-range correlations in texts have been related to large scales characteristics of the novels such as the story being told, the style of the book, the author, and the language (1, 2, 16-21). However, the mechanisms explaining these connections are still missing (see ref. 2 for a recent proposal). Without such mechanisms, many fundamental questions cannot be answered. For instance, why all previous investigations observed long-range correlations despite their radically different approaches? How and which correlations can flow from the high-level semantic structures down to the crude symbolic sequence in the presence of so many arbitrary influences? What information is gained on the large structures by looking at smaller ones? Finally, what is the origin of the longrange c...

We introduce a cycle-expansion (fully deterministic) technique to compute the asymptotic behavior of arbitrary order transport moments. The theory is applied to different kinds of one-dimensional intermittent maps, and Lorentz gas with infinite horizon, confirming the typical appearance of phase transitions in the transport spectrum.

We show that the nontwist phenomena previously observed in Hamiltonian systems exist also in time-reversible non-Hamiltonian systems. In particular, we study the two standard collision-reconnection scenarios and we compute the parameter space breakup diagram of the shearless torus. Besides the Hamiltonian routes, the breakup may occur due to the onset of attractors. We study these phenomena in coupled phase oscillators and in non-area-preserving maps.

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.

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.Mathematics Subject Classification (2010): 60G50, 60F05 (82C41, 60G55).

We study diffusion on a periodic billiard table with an infinite horizon in the limit of narrow corridors. An effective trapping mechanism emerges according to which the process can be modeled by a Lévy walk combining exponentially distributed trapping times with free propagation along paths whose precise probabilities we compute. This description yields an approximation of the mean squared displacement of infinite-horizon billiards in terms of two transport coefficients, which generalizes to this anomalous regime the Machta-Zwanzig approximation of normal diffusion in finite-horizon billiards [J. Machta and R. Zwanzig, Phys. Rev. Lett. 50, 1959 (1983)PRLTAO0031-900710.1103/PhysRevLett.50.1959].

The Sparre-Andersen theorem is a remarkable result in one-dimensional random walk theory concerning the universality of the ubiquitous first-passage-time distribution. It states that the probability distribution ρn of the number of steps needed for a walker starting at the origin to land on the positive semi-axes does not depend on the details of the distribution for the jumps of the walker, provided this distribution is symmetric and continuous, where in particular ρn ∼ n −3/2 for large number of steps n. On the other hand, there are many physical situations in which the time spent by the walker in doing one step depends on the length of the step and the interest concentrates on the time needed for a return, not on the number of steps. Here we modify the Sparre-Andersen proof to deal with such cases, in rather general situations in which the time variable correlates with the step variable. As an example we present a natural process in 2D that shows deviations from normal scaling are present for the first-passage-time distribution on a semi plane. For more than a century (see, for instance [1]) random walks have played a crucial role as a theoretical tool to model an impressive number of physical (and not only) problems. Fundamental questions in the theory of stochastic processes are related to the problem of when a variable in a system enters some a priori specified state for the first time: the first-passage-time. Knowledge of the first-passage-time distribution (FPTD) finds application in many diverse areas of the natural sciences and economics; from the spike distribution in neuronal dynamics, the meeting time of two molecules in diffusion-controlled chemical reactions, the cluster density in aggregation reactions, to the price of a stock reaching a specific value and the ruin problem of actuarial science (see [2] for an extensive treatment of the problem, and early references to applications); in the last decades, relevance of such a theory to non-equilibrium problems has been exploited [3]. In this framework the Sparre-Andersen theorem (SA) [4][5][6] plays an outstanding role: in physics it has been invoked in the study of persistence in stochastic spin models and random walks [7] the study of polymer dynamics [8] and in the analysis of scattering from a Lorentz slab [9]. In particular, SA states that the probability that a random walker who starts at the origin, enters the positive semi-axis for the first time (its first-passage-time) after n steps is independent of the particular details of the jump length distribution, provided that it is symmetric about the origin and continuous: in such a case the decay of the FPTD has the universal asymptotics n −3/2 . Here the conceptual import of SA is apparent: it provides an outstanding example of universality in the realm of stochastic processes, with a huge universality class. The origin of such universality is subtler than other examples in probability: to our knowledge it cannot be encompassed by simple renormalization, like the central limit theorem (see, fo...