The Totally Asymmetric Simple Exclusion Process (TASEP) is often considered one of the fundamental models of non-equilibrium statistical mechanics, due to its well understood steady state and the fact that it can exhibit condensation, phase separation and phase transitions in one spatial dimension. As a minimal model of traffic flow it has enjoyed many applications, including the transcription of proteins by ribosomal motors moving along an mRNA track, the transport of cargo between cells and more human-scale traffic flow problems such as the dynamics of bus routes. It consists of a one-dimensional lattice of sites filled with a number of particles constrained to move in a particular direction, which move to adjacent sites probabilistically and interact by mutual exclusion. The study of non-Markovian interacting particle systems is in its infancy, due in part to a lack of a framework for addressing them analytically. In this thesis we extend the TASEP to allow the rate of transition between sites to depend on how long the particle in question has been stationary by using non-Poissonian waiting time distributions. We discover that if the waiting time distribution has infinite variance, a dynamic condensation effect occurs whereby every particle on the system comes to rest in a single traffic jam. As the lattice size increases, so do the characteristic condensate lifetimes and the probability that a condensate will interact with the preceding one by forming out of its remnants. This implies that the thermodynamic limit depends on the dynamics of such spatially complete condensates. As the characteristic condensate lifetimes increase, the standard continuous time Monte Carlo simulation method results in an increasingly large fraction of failed moves. This is computationally costly and led to a limit on the sizes of lattice we could simulate. We integrate out the failed moves to create a rejection-free algorithm which allows us to see the interacting condensates more clearly. We find that if condensates do not fully dissolve, the condensate lifetime ages and saturates to a particular value. An unforeseen consequence of this new technique, is that it also allowed us to gain a mathematical understanding of the ageing of condensates, and its dependence on system size. Using this we can see that the fraction of time spent in the spatially complete condensate tends to one in the thermodynamic limit. A random walker in a random force field has to escape potential wells of random depth, which gives rise to a power law waiting time distribution. We use the non-Markovian TASEP to investigate this model with a number of interacting particles. We find that if the potential well is re-sampled after every failed move, then this system is equivalent to the non-Markovian TASEP. If the potential well is only re-sampled after a successful move, then we restore particle-hole symmetry, allow condensates to completely dissolve, and the thermodynamic limit spends a finite fraction of time in the spatially complete state. We then generalised the non-Markovian TASEP to allow for particles to move in both directions. We find that the full condensation effect remains robust except for the case of perfect symmetry.

The analysis of large disordered complex networks has recently received enormous attention motivated by both academic and commercial interest. The most important results in this discipline have come from the analysis of stochastic models which mimic the growth and evolution of real networks as they change over time. The purpose of this thesis is to introduce various novel processes which dictate the development of a network on a small scale, and use techniques learned from statistical physics to derive the dynamical and structural properties of the network on the macroscopic scale. We introduce each model as a set of mechanisms determining how a network changes over a small period in time, from these rules we derive several topological properties of the network after many iterations, most notably the degree distribution. 1. In the rst mechanism, nodes are introduced and linked to older nodes in the network in such a way as to create triangles and maintain a high level of clustering. The mechanism resembles the growth of a citation network and we demonstrate analytically that the mechanism introduced su ces to explain the power-law form commonly found in citation distributions. 2. The second mechanism involves edge rewiring processes - detaching one end of an edge and reattaching it, either to a random node anywhere in the network or to one selected locally. 3. We analyse a variety of processes based around a novel fragmentation mechanism. 4. The nal model concerns the problem of nding the electrical resistance across a network. The network grows as a random tree, as it grows the distribution of resistance converges towards a steady state solution. We nd an application of the relatively recent concept of a random Fibonacci sequence in deriving the rate of convergence of the mean.