Sums of correlated exponentials: two types of Gaussian correlation structures

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 non-hierarchically strongly correlated random variables which is based on the partition function of the Sherrington-Kirkpatrick (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 GREM-inspired processes and Ruelle’s probability cascades. For this purpose, an abstract quenched large deviations principle of the G¨artner-Ellis 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 well-known properties of Ruelle’s probability cascades and the Bolthausen-Sznitman coalescent. We study the properties of the multidimensional Parisi functional by establishing a link with a certain class of semi-linear 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 coarse-graining 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 coarse-grained 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.