Abstract
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 powerlaw 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 socalled 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 powerlaw random
graphs and the antiferromagnetic Potts model on the Erd¿osRényi 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 powerlaw
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 powerlaw exponent
and not on any other detail of the degree distribution.
The proofs rely on the locally treelike 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 powerlaw exponents 5 a more detailed analysis is necessary.
We also study the case where the powerlaw exponent 2 (1, 2) for which the mean
degree is infinite and the graph is no longer locally treelike. 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 hightemperature 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.
Original language  English 

Qualification  Doctor of Philosophy 
Awarding Institution 

Supervisors/Advisors 

Award date  26 Mar 2013 
Place of Publication  Eindhoven 
Publisher  
Print ISBNs  9789038633480 
DOIs  
Publication status  Published  2013 
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Cite this
Dommers, S. (2013). Spin models on random graphs. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR750979