Zufallsgraphen sind Graphen, die durch einen zufälligen Prozess erzeugt werden. Ein im Zusammenhang mit Zufallsgraphen häufig auftretendes Phänomen ist, dass sich die typischen Eigenschaften eines Graphen durch Hinzufügen einer relativ kleinen Anzahl von zufälligen Kanten radikal verändern. Wir betrachten den Zufallsgraphen G(n,p), der n Knoten enthält und in dem zwei Knoten unabhängig und mit Wahrscheinlichkeit p durch eine Kante verbunden sind. Erdös und Rényi zeigten, dass ein Graph für p = c/n und c < 1 mit hoher Wahrscheinlichkeit aus Komponenten mit O(log n) Knoten besteht. Für p = c/n und c > 1 enthält G(n,p) mit hoher Wahrscheinlichkeit genau eine Komponente mit Theta(n) Knoten, welche viel größer als alle anderen Komponenten ist. Dieser Punkt in der Entwicklung des Graphen, an dem sich die Komponentenstruktur durch eine kleine Erhöhung der Anzahl von Kanten stark verändert, wird Phasenübergang genannt. Wenn p = (1+epsilon)/n, wobei epsilon eine Funktion von n ist, die gegen 0 geht, sind wir in der kritischen Phase, welche eine der interessantesten Phasen der Entwicklung des Zufallsgraphen ist. In dieser Arbeit betrachten wir drei verschiedene Modelle von Zufallsgraphen. In Kapitel 4 studieren wir den Minimalgrad-Graphenprozess. In diesem Prozess werden sukzessive Kanten vw hinzugefügt, wobei v ein zuällig ausgewählter Knoten von minimalem Grad ist. Wir beweisen, dass es in diesem Graphenprozess einen Phasenübergang, und wie im G(n,p) einen Doppelsprung, gibt. Die zwei anderen Modelle sind Zufallsgraphen mit einer vorgeschriebenen Gradfolge und zufällige gerichtete Graphen. Für diese Modelle wurde bereits in den Arbeiten von Molloy und Reed (1995), Karp (1990) und Luczak (1990) gezeigt, dass es einen Phasenübergang bezüglich der Komponentenstruktur gibt. In dieser Arbeit untersuchen wir in Kapitel 5 und 6 die kritische Phase dieser Prozesse genauer, und zeigen, dass sich diese Modelle ähnlich zum G(n,p) verhalten. / Random graphs are graphs which are created by a random process. A common phenomenon in random graphs is that the typical properties of a graph change radically by the addition of a relatively small number of random edges. This phenomenon was first investigated in the seminal papers of Erdös and Rényi. We consider the graph G(n,p) which contains n vertices, and where any two vertices are connected by an edge independently with probability p. Erdös and Rényi showed that if p = c/n$ and c < 1, then with high probability G(n,p) consists of components with O(log n) vertices. If p = c/n$ and c>1, then with high probability G(n,p) contains exactly one component, called the giant component, with Theta(n) vertices, which is much larger than all other components. The point at which the giant component is formed is called the phase transition. If we let $p = (1+epsilon)/n$, where epsilon is a function of n tending to 0, we are in the critical phase of the random graph, which is one of the most interesting phases in the evolution of the random graph. In this case the structure depends on how fast epsilon tends to 0. In this dissertation we consider three different random graph models. In Chapter 4 we consider the so-called minimum degree graph process. In this process edges vw are added successively, where v is a randomly chosen vertex with minimum degree. We prove that a phase transition occurs in this graph process as well, and also that it undergoes a double jump, similar to G(n,p). The two other models we will consider, are random graphs with a given degree sequence and random directed graphs. In these models the point of the phase transition has already been found, by Molloy and Reed (1995), Karp (1990) and Luczak (1990). In Chapter 5 and 6 we investigate the critical phase of these processes, and show that their behaviour resembles G(n,p).

Using the AdS/CFT correspondence, we study the anisotropic transport properties of both supersymmetric and non-supersymmetric matter fields on (2+1)-dimensional defects coupled to a (3+1)-dimensional N=4 SYM "heat bath". We address on the one hand the purely conformal defect where the only non-vanishing background field that we turn on is a "topological", parameter parametrizing the impact on the bulk. On the other hand we also address the case of a finite external background magnetic field, finite net charge density and finite mass. We find in the purely conformal limit that the system possesses a conduction threshold given by the wave number of the perturbation and that the charge transport arises from a quasiparticle spectrum which is consistent with an intuitive picture where the defect acquires a finite width in the direction of the SYM bulk. We also examine finite-coupling modifications arising from higher derivative interactions in the probe brane action. In the case of finite density, mass and magnetic field, our results generalize the conformal case. We discover at high frequencies a spectrum of quasiparticle resonances due to the magnetic field and finite density and at small frequencies a Drude-like expansion around the DC limit. Both of these regimes display many generic features and some features that we attribute to strong coupling, such as a minimum DC conductivity and an unusual behavior of the "cyclotron" and plasmon frequencies, which become correlated to the resonances found in the conformal case. We further study the hydrodynamic regime and the relaxation properties, in which the system displays a set of different possible transitions to the collisionless regime. The mass dependence can be cast in two regimes: a generic relativistic behavior dominated by the UV and a non-linear hydrodynamic behavior dominated by the IR. In the massless case, we also extend earlier results to find an interesting duality under the transformation of the conductivity and the exchange of density and magnetic field. Furthermore, we look at the thermodynamics and the phase diagram, which reproduces general features found earlier in 3+1 dimensional systems and demonstrates stability in the relevant phase.

Using the AdS/CFT correspondence, we study the anisotropic transport properties of both supersymmetric and non-supersymmetric matter fields on (2+1)-dimensional defects coupled to a (3+1)-dimensional N=4 SYM "heat bath". We address on the one hand the purely conformal defect where the only non-vanishing background field that we turn on is a "topological", parameter parametrizing the impact on the bulk. On the other hand we also address the case of a finite external background magnetic field, finite net charge density and finite mass. We find in the purely conformal limit that the system possesses a conduction threshold given by the wave number of the perturbation and that the charge transport arises from a quasiparticle spectrum which is consistent with an intuitive picture where the defect acquires a finite width in the direction of the SYM bulk. We also examine finite-coupling modifications arising from higher derivative interactions in the probe brane action. In the case of finite density, mass and magnetic field, our results generalize the conformal case. We discover at high frequencies a spectrum of quasiparticle resonances due to the magnetic field and finite density and at small frequencies a Drude-like expansion around the DC limit. Both of these regimes display many generic features and some features that we attribute to strong coupling, such as a minimum DC conductivity and an unusual behavior of the "cyclotron" and plasmon frequencies, which become correlated to the resonances found in the conformal case. We further study the hydrodynamic regime and the relaxation properties, in which the system displays a set of different possible transitions to the collisionless regime. The mass dependence can be cast in two regimes: a generic relativistic behavior dominated by the UV and a non-linear hydrodynamic behavior dominated by the IR. In the massless case, we also extend earlier results to find an interesting duality under the transformation of the conductivity and the exchange of density and magnetic field. Furthermore, we look at the thermodynamics and the phase diagram, which reproduces general features found earlier in 3+1 dimensional systems and demonstrates stability in the relevant phase.