Spin models on random graphs

In the past decades complex networks and their behavior have attracted much attention. In the real world many of such networks can be found, for instance as social, information, technological and biological networks. An interesting property that many of them share is that they are scale free. Such networks have many nodes with a moderate amount of links, but also a significant amount of nodes with a very high number of links. The latter type of nodes are called hubs and play an important role in the behavior of the network. To model scale free networks, we use power-law random graphs. This means that their degree sequences obey a power law, i.e., the fraction of vertices that have k neighbors is proportional to k- for some > 1. Not only the structure of these networks is interesting, also the behavior of processes living on these networks is a fascinating subject. Processes one can think of are opinion formation, the spread of information and the spread of viruses. It is especially interesting if these processes undergo a so-called phase transition, i.e., a minor change in the circumstances suddenly results in completely different behavior. Hubs in scale free networks again have a large influence on processes living on them. The relation between the structure of the network and processes living on the network is the main topic of this thesis. We focus on spin models, i.e., Ising and Potts models. In physics, these are traditionally used as simple models to study magnetism. When studied on a random graph, the spins can, for example, be considered as opinions. In that case the ferromagnetic or antiferromagnetic interactions can be seen as the tendency of two connected persons in a social network to agree or disagree, respectively. In this thesis we study two models: the ferromagnetic Ising model on power-law random graphs and the antiferromagnetic Potts model on the Erd¿os-Rényi random graph. For the first model we derive an explicit formula for the thermodynamic limit of the pressure, generalizing a result of Dembo and Montanari to random graphs with power-law exponent > 2, for which the variance of degrees is potentially infinite. We furthermore identify the thermodynamic limit of the magnetization, internal energy and susceptibility. For this same model, we also study the phase transition. We identify the critical temperature and compute the critical exponents of the magnetization and susceptibility. These exponents are universal in the sense that they only depend on the power-law exponent and not on any other detail of the degree distribution. The proofs rely on the locally tree-like structure of the random graph. This means that the local neighborhood of a randomly chosen vertex behaves like a branching process. Correlation inequalities are used to show that it suffices to study the behavior of the Ising model on these branching processes to obtain the results for the random graph. To compute the critical temperature and critical exponents we derive upper and lower bounds on the magnetization and susceptibility. These bounds are essentially Taylor approximations, but for power-law exponents 5 a more detailed analysis is necessary. We also study the case where the power-law exponent 2 (1, 2) for which the mean degree is infinite and the graph is no longer locally tree-like. We can, however, still say something about the magnetization of this model. For the antiferromagnetic Potts model we use an interpolation scheme to show that the thermodynamic limit exists. For this model the correlation inequalities do not hold, thus making it more difficult to study. We derive an extended variational principle and use to it give upper bounds on the pressure. Furthermore, we use a constrained secondmoment method to show that the high-temperature solution is correct for high enough temperature. We also show that this solution cannot be correct for low temperatures by showing that the entropy becomes negative if it were to be correct, thus identifying a phase transition

Spin models on random graphs

In the past decades complex networks and their behavior have attracted much attention. In the real world many of such networks can be found, for instance as social, information, technological and biological networks. An interesting property that many of them share is that they are scale free. Such networks have many nodes with a moderate amount of links, but also a significant amount of nodes with a very high number of links. The latter type of nodes are called hubs and play an important role in the behavior of the network. To model scale free networks, we use power-law random graphs. This means that their degree sequences obey a power law, i.e., the fraction of vertices that have k neighbors is proportional to k- for some > 1. Not only the structure of these networks is interesting, also the behavior of processes living on these networks is a fascinating subject. Processes one can think of are opinion formation, the spread of information and the spread of viruses. It is especially interesting if these processes undergo a so-called phase transition, i.e., a minor change in the circumstances suddenly results in completely different behavior. Hubs in scale free networks again have a large influence on processes living on them. The relation between the structure of the network and processes living on the network is the main topic of this thesis. We focus on spin models, i.e., Ising and Potts models. In physics, these are traditionally used as simple models to study magnetism. When studied on a random graph, the spins can, for example, be considered as opinions. In that case the ferromagnetic or antiferromagnetic interactions can be seen as the tendency of two connected persons in a social network to agree or disagree, respectively. In this thesis we study two models: the ferromagnetic Ising model on power-law random graphs and the antiferromagnetic Potts model on the Erd¿os-Rényi random graph. For the first model we derive an explicit formula for the thermodynamic limit of the pressure, generalizing a result of Dembo and Montanari to random graphs with power-law exponent > 2, for which the variance of degrees is potentially infinite. We furthermore identify the thermodynamic limit of the magnetization, internal energy and susceptibility. For this same model, we also study the phase transition. We identify the critical temperature and compute the critical exponents of the magnetization and susceptibility. These exponents are universal in the sense that they only depend on the power-law exponent and not on any other detail of the degree distribution. The proofs rely on the locally tree-like structure of the random graph. This means that the local neighborhood of a randomly chosen vertex behaves like a branching process. Correlation inequalities are used to show that it suffices to study the behavior of the Ising model on these branching processes to obtain the results for the random graph. To compute the critical temperature and critical exponents we derive upper and lower bounds on the magnetization and susceptibility. These bounds are essentially Taylor approximations, but for power-law exponents 5 a more detailed analysis is necessary. We also study the case where the power-law exponent 2 (1, 2) for which the mean degree is infinite and the graph is no longer locally tree-like. We can, however, still say something about the magnetization of this model. For the antiferromagnetic Potts model we use an interpolation scheme to show that the thermodynamic limit exists. For this model the correlation inequalities do not hold, thus making it more difficult to study. We derive an extended variational principle and use to it give upper bounds on the pressure. Furthermore, we use a constrained secondmoment method to show that the high-temperature solution is correct for high enough temperature. We also show that this solution cannot be correct for low temperatures by showing that the entropy becomes negative if it were to be correct, thus identifying a phase transition

Ising critical exponents on random trees and graphs

We study the critical behavior of the ferromagnetic Ising model on random trees as well as so-called locally tree-like random graphs. We pay special attention to trees and graphs with a power-law offspring or degree distribution whose tail behavior is characterized by its power-law exponent $\tau>2$. We show that the critical temperature of the Ising model equals the inverse hyperbolic tangent of the inverse of the mean offspring or mean forward degree distribution. In particular, the inverse critical temperature equals zero when $\tau\in(2,3]$ where this mean equals infinity. We further study the critical exponents $\delta, \beta$ and $\gamma$, describing how the (root) magnetization behaves close to criticality. We rigorously identify these critical exponents and show that they take the values as predicted by Dorogovstev, et al. and Leone et al. These values depend on the power-law exponent $\tau$, taking the mean-field values for $\tau>5$, but different values for $\tau\in(3,5)$

Berry-Esseen bounds in the inhomogeneous Curie-Weiss model with external field

We study the inhomogeneous Curie-Weiss model with external field, where the inhomo-geneity is introduced by adding a positive weight to every vertex and letting the interaction strength between two vertices be proportional to the product of their weights. In this model, the sum of the spins obeys a central limit theorem outside the critical line. We derive a Berry-Esseen rate of convergence for this limit theorem using Stein's method for exchangeable pairs. For this, we, amongst others, need to generalize this method to a multidimensional setting with unbounded random variables

Scattering processes in quantum dot devices with the example of an amplifier

All experiments performed in this work deal with a device that is a quantum dot based semiconductor optical amplifier. This chapter explains why especially InAs quantum dots in a waveguide were chosen to perform them. In addition all basics necessary to understand the experimental methods, the setup and the models will be described. The first section will deal with semiconductors and the special terminology used for describing them. After that the advantages of quantum dots in comparison to other semiconductor nanostructures are clarified. In the third section the device itself is focused, the Semiconductor Optical Amplifier (SOA) and its characteristics. First the design will be discussed then the optical characteristics. The performed experiments are described in detail in the fourth section of this chapter. Both the technical setup and the used methods will be explained. This sections is completed with a first characterization of the sample. At last I introduce the theory necessary for following the models used in chapter 4 and 5 of this thesis

Metastability in the reversible inclusion process

We study the condensation regime of the finite reversible inclusion process, i.e., the inclusion process on a finite graph S with an underlying random walk that admits a reversible measure. We assume that the random walk kernel is irreducible and its reversible measure takes maximum value on a subset of vertices S⋆⊆S. We consider initial conditions corresponding to a single condensate that is localized on one of those vertices and study the metastable (or tunneling) dynamics. We find that, if the random walk restricted to S⋆ is irreducible, then there exists a single time-scale for the condensate motion. In this case we compute this typical time-scale and characterize the law of the (properly rescaled) limiting process. If the restriction of the random walk to S⋆ has several connected components, a metastability scenario with multiple time-scales emerges. We prove such a scenario, involving two additional time-scales, in a one-dimensional setting with two metastable states and nearest-neighbor jumps

Metastability for Glauber dynamics on random graphs

Analysis and Stochastic

A Two-populations Ising model on diluted Random Graphs

We consider the Ising model for two interacting groups of spins embedded in an Erd\"{o}s-R\'{e}nyi random graph. The critical properties of the system are investigated by means of extensive Monte Carlo simulations. Our results evidence the existence of a phase transition at a value of the inter-groups interaction coupling $J_{12}^C$ which depends algebraically on the dilution of the graph and on the relative width of the two populations, as explained by means of scaling arguments. We also measure the critical exponents, which are consistent with those of the Curie-Weiss model, hence suggesting a wide robustness of the universality class.Comment: 11 pages, 4 figure