Non-homogeneous random walks: from the transport properties to the statistics of occupation times

This dissertation details our research on random walks seen as simple mathematical models useful to describe the complex dynamics of many physical systems. In particular, we focus on the role of spatial inhomogeneity in determining the deviations from the standard behaviour. In the first chapter we present a general method that can be used to obtain the continuum limit for the evolution equations of a random walk with nearest neighbour jumps, from which one can derive the asymptotic properties and deduce the physical interpretation of the walk itself. Then in the following we adopt this method to discuss two particular models. The first model, which we call Gillis random walk, is treated in the second chapter and consists in a random walk with space-dependent drift. Although lacking translational invariance, it provides one of the very few examples of a stochastic system allowing for a number of exact results. From the continuum limit, one deduces that this model provides a microscopical description for the problem of Brownian diffusion in a logarithmic potential, and indeed we compare the results regarding the diffusion problem already present in the literature with the behaviour of the Gillis random walk, finding good agreement. The second model, which we have originally introduced in the literature and deal with in the third chapter, is a correlated random walk closely related to the Lévy-Lorentz gas, a stochastic system where a particle is scattered by static points arranged on a line in such a way that the distances between first neighbour scatterers are independent and identically distributed random variables, drawn from a heavy-tailed distribution. Our model results from a particular procedure of average over all possible arrangements of scatterers and it is mathematically described as a correlated random walk on the integer lattice, where at each step the particle can be either reflected or transmitted according to a space-dependent probability. We apply the continuum limit and derive the long-time properties of the system, which to some extent match those of the original Lévy-Lorentz gas. In the fourth and last chapter we consider the problem of occupation times for onedimensional random walks, showing that for a wide class of processes a single exponent related to a local property of the system is sufficient to describe the distributions of the variables of interests. We test our findings using the two stochastic models presented in the previous chapters and obtain good agreement with our theory. However, our result breaks down, for example, if we consider continuous time random walk models in which the distribution of waiting times between steps does not possess finite mean. Nevertheless, we show how also in this case the theory developed in the first part of the chapter is useful to obtain the statistics of occupation times. We revise some of the results already present in the literature in terms of our theory and test the results on a novel continuous time model based on the dynamics of the Gillis random walk, finding good agreement with both the literature and our theory.