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

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

Ising models on power-law random graphs

We study a ferromagnetic Ising model on random graphs with a power-law degree distribution and compute the thermodynamic limit of the pressure when the mean degree is finite (degree exponent $\tau>2$), for which the random graph has a tree-like structure. For this, we adapt and simplify an analysis by Dembo and Montanari, which assumes finite variance degrees ($\tau>3$). We further identify the thermodynamic limits of various physical quantities, such as the magnetization and the internal energy

Antiferromagnetic Potts model on the Erdos-Renyi random graph

We study the antiferromagnetic Potts model on the Poissonian Erd\"os-R\'enyi random graph. By identifying a suitable interpolation structure and an extended variational principle, together with a positive temperature second-moment analysis we prove the existence of a phase transition at a positive critical temperature. Upper and lower bounds on the temperature critical value are obtained from the stability analysis of the replica symmetric solution (recovered in the framework of Derrida-Ruelle probability cascades)and from a positive entropy argument.Comment: 36 pages, revisions to improve resul

Partition function zeros for the Ising model on complete graphs and on annealed scale-free networks

We analyze the partition function of the Ising model on graphs of two different types: complete graphs, wherein all nodes are mutually linked and annealed scale-free networks for which the degree distribution decays as $P(k)\sim k^{-\lambda}$. We are interested in zeros of the partition function in the cases of complex temperature or complex external field (Fisher and Lee-Yang zeros respectively). For the model on an annealed scale-free network, we find an integral representation for the partition function which, in the case $\lambda > 5$, reproduces the zeros for the Ising model on a complete graph. For $3<\lambda < 5$ we derive the $\lambda$-dependent angle at which the Fisher zeros impact onto the real temperature axis. This, in turn, gives access to the $\lambda$-dependent universal values of the critical exponents and critical amplitudes ratios. Our analysis of the Lee-Yang zeros reveals a difference in their behaviour for the Ising model on a complete graph and on an annealed scale-free network when $3<\lambda <5$. Whereas in the former case the zeros are purely imaginary, they have a non zero real part in latter case, so that the celebrated Lee-Yang circle theorem is violated.Comment: 36 pages, 31 figure

Nonlinear optics and saturation behavior of quantum dot samples under continuous wave driving

The nonlinear optical response of self-assembled quantum dots is relevant to the application of quantum dot based devices in nonlinear optics, all-optical switching, slow light and self-organization. Theoretical investigations are based on numerical simulations of a spatially and spectrally resolved rate equation model, which takes into account the strong coupling of the quantum dots to the carrier reservoir created by the wetting layer states. The complex dielectric susceptibility of the ground state is obtained. The saturation is shown to follow a behavior in between the one for a dominantly homogeneously and inhomogeneously broadened medium. Approaches to extract the nonlinear refractive index change by fringe shifts in a cavity or self-lensing are discussed. Experimental work on saturation characteristic of InGa/GaAs quantum dots close to the telecommunication O-band (1.24-1.28 mm) and of InAlAs/GaAlAs quantum dots at 780 nm is described and the first demonstration of the cw saturation of absorption in room temperature quantum dot samples is discussed in detail