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...

We use random matrix theory to study the spectrum of random geometric graphs, a fundamental model of spatial networks. Considering ensembles of random geometric graphs we look at short range correlations in the level spacings of the spectrum via the nearest neighbour and next nearest neighbour spacing distribution and long range correlations via the spectral rigidity ∆ 3 statistic. These correlations in the level spacings give information about localisation of eigenvectors, level of community structure and the level of randomness within the networks. We find a parameter dependent transition between Poisson and Gaussian orthogonal ensemble statistics. That is the spectral statistics of spatial random geometric graphs fits the universality of random matrix theory found in other models such as Erdős-Rényi, Barabási-Albert and Watts-Strogatz random graphs.

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.

We analyse deterministic diffusion in a simple, one-dimensional setting consisting of a family of four parameter dependent, chaotic maps defined over the real line. When iterated under these maps, a probability density function spreads out and one can define a diffusion coefficient. We look at how the diffusion coefficient varies across the family of maps and under parameter variation. Using a technique by which Taylor-Green-Kubo formulae are evaluated in terms of generalised Takagi functions, we derive exact, fully analytical expressions for the diffusion coefficients. Typically, for simple maps these quantities are fractal functions of control parameters. However, our family of four maps exhibits both fractal and linear behavior. We explain these different structures by looking at the topology of the Markov partitions and the ergodic properties of the maps.

We investigate three different methods for systematically approximating the diffusion coefficient of a deterministic random walk on the line which contains dynamical correlations that change irregularly under parameter variation. Capturing these correlations by incorporating higher order terms, all schemes converge to the analytically exact result. Two of these methods are based on expanding the Taylor-Green-Kubo formula for diffusion, whilst the third method approximates Markov partitions and transition matrices by using a slight variation of the escape rate theory of chaotic diffusion. We check the practicability of the different methods by working them out analytically and numerically for a simple one-dimensional map, study their convergence and critically discuss their usefulness in identifying a possible fractal instability of parameter-dependent diffusion, in case of dynamics where exact results for the diffusion coefficient are not available.