Samenvatting
In this thesis, we study the limiting behaviour of the large sums of strongly correlated exponentials
as the number of their summands and the effective dimension of the correlation structure
simultaneously tend to infinity.We consider two types of such sums which are generated by two
a priori very different Gaussian correlation structures. The first type is a sum of hierarchically
correlated random variables which is based on the partition function of Derrida’s generalised
random energy model (GREM) with external field. The second type is an infinitesimal sum
of genuinely nonhierarchically strongly correlated random variables which is based on the
partition function of the SherringtonKirkpatrick (SK) model with multidimensional spins. We
consider the asymptotic behaviour (the thermodynamic limit) of these two sums on a logarithmic
scale (i.e., at the level of free energy) and also at a more refined level of their fluctuations
(i.e., at the level of weak limiting laws). Interestingly for the SK model with multidimensional
spins, we find traces of a hierarchical organisation in the thermodynamic limit. This supports
the conjectured in theoretical physics universal behaviour of the sums of such sort.
Concerning the SK model with multidimensional spins, we obtain the following results.
We prove upper and lower bounds on the free energy of this model in terms of variational
inequalities. The bounds are based on a multidimensional extension of the Parisi functional.We
generalise and unify the comparison scheme of Aizenman, Sims and Starr and the one of Guerra
involving the GREMinspired processes and Ruelle’s probability cascades. For this purpose, an
abstract quenched large deviations principle of the G¨artnerEllis type is obtained. We derive
Talagrand’s representation of Guerra’s remainder term for the SK model with multidimensional
spins. The derivation is based on wellknown properties of Ruelle’s probability cascades and
the BolthausenSznitman coalescent. We study the properties of the multidimensional Parisi
functional by establishing a link with a certain class of semilinear partial differential equations.
We embed the problem of strict convexity of the Parisi functional in a more general setting
and prove the convexity in some particular cases which, however, do not cover the original
setup of Talagrand. Finally, we prove the Parisi formula for the local free energy in the case of
multidimensional Gaussian a priori distribution of spins using Talagrand’s methodology of a
priori estimates.
Concerning the GREM in the presence of uniform external field, we obtain the following
results. We compute the fluctuations of the ground state and of the partition function in the
thermodynamic limit for all admissible values of parameters. We find that the fluctuations are
described by a hierarchical structure which is obtained by a certain coarsegraining of the initial
hierarchical structure of the GREM with external field. We provide an explicit formula for
the free energy of the model. We also derive some large deviation results providing an expression
for the free energy in a class of models with Gaussian Hamiltonians and external field.
Finally, we prove that the coarsegrained parts of the system emerging in the thermodynamic
limit tend to have a certain optimal magnetisation, as prescribed by strength of external field
and by parameters of the GREM.
Originele taal2  Engels 

Kwalificatie  Doctor in de Filosofie 
Toekennende instantie 

Begeleider(s)/adviseur 

Datum van toekenning  24 jun 2008 
Plaats van publicatie  Berlin 
Uitgever  
Status  Gepubliceerd  2008 
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Citeer dit
Klymovskiy, A. (2008). Sums of correlated exponentials: two types of Gaussian correlation structures. TU Berlin.